Abstract
Friedlander, Macêdo, and Pong recently introduced the projected polar proximal point algorithm (P4A) for solving optimization problems by using the closed perspective transforms of convex objectives. We analyse a generalization (GP4A) which replaces the closed perspective transform with a more general closed gauge. We decompose GP4A into the iterative application of two separate operators, and analyse it as a splitting method. By showing that GP4A and its under-relaxations exhibit global convergence whenever a fixed point exists, we obtain convergence guarantees for P4A by letting the gauge specify to the closed perspective transform for a convex function. We then provide easy-to-verify sufficient conditions for the existence of fixed points for the GP4A, using the Minkowski function representation of the gauge. Conveniently, the approach reveals that global minimizers of the objective function for P4A form an exposed face of the dilated fundamental set of the closed perspective transform.
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1 Introduction
Friedlander, Macêdo, and Pong introduced the projected polar proximal point algorithm (\({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\), Definition 4) as the first proximal-point-like algorithm based on the polar envelope of a gauge [6]. The motivation to study such algorithms stems from the polar envelope’s relationship to infimal max convolution; Friedlander, Macêdo, and Pong showed that this relationship is analogous to the connection between the Moreau envelope and infimal convolution. They also illuminated useful variational properties of a duality framework admitted by such problems [5]. It is these special properties of the algorithm, and the rich associated theoretical framework that motivated its construction, that make its global convergence an interesting question.
The method makes use of the closed perspective transform for a proper convex function \(f:X \rightarrow {{\mathbb {R}}}\):
Here \(f_\infty \) denotes the recession function of f [1, Definition 2.5.1], which satisfies
and \(\mathrm{epi}h\) denotes the epigraph of a proper convex function h and \(C_\infty \) denotes the recession cone of a set C (see, for example, [8, Chapter 6]). The perspective \(f^\pi \) is proper closed convex [7, Page 67] and is characterized by
Friedlander, Macêdo, and Pong also provided a result [6, Theorem 7.5] showing convergence under an assumption of strong convexity of \((f^\pi )^2\); however, the question of convergence more generally has remained open until now.
1.1 Outline and contributions
In Sect. 2, we recall familiar notation and concepts from convex analysis. In Sect. 3, we recall and analyse the polar proximity operator, a fundamental component of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\). In particular, we show that it is firmly quasinonexpansive (Theorem 3.4).
In Sect. 4, we recall the algorithm \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) and introduce a generalization thereof, \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) (Definition 5). Our motivation in so doing is that \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) may be described and studied as a 2-operator splitting method, a flexibility afforded by its definition on the lifted space \(X \times {{\mathbb {R}}}\). In Sect. 4.1, we exploit this flexibility to show that, when the operator associated to \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) has a nonempty fixed point set, its fixed points all share a special property (Proposition 4.4), on which our analysis depends. In Sect. 4.2, we use this property to show that the operator is strictly quasinonexpansive (Theorem 4.8) and admits global convergence of sequences to a fixed point, whenever one exists (Theorem 4.11). We show similar results for the algorithm’s under-relaxed variants (Theorem 4.10). In Sect. 4.3, we provide convergence results for the associated shadow sequences. In Sect. 4.4, we provide an example that shows the operator associated with \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) is not, generically, firmly quasinonexpansive.
In Sect. 5, we provide sufficient conditions to guarantee fixed points of \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) (Theorem 5.2). Moreover, when \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) specifies to \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\), we show that set of global minimizers of f defines an exposed face of the fundamental set for the perspective function \(f^\pi \) (Theorem 5.3). The latter results connect to known results ( [6, Theorem 7.4]) about fixed points of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\), and we explain how (Remark 2).
In Sect. 6, we connect our sufficient conditions for fixed point existence, with our convergence results that depend on that existence, to state a simple global convergence guarantee (Theorem 6.1). It shows that \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) is globally convergent in the full generality of [6].
2 Preliminaries
Throughout, X is a finite dimensional Euclidean space with the Euclidean norm, and we will work extensively with the space \(X \times {{\mathbb {R}}}\). For ease of clarity, when we work with a 2-tuple \((y,\lambda ) \in X \times {{\mathbb {R}}}\), it should be understood that \(y \in X\) and \(\lambda \in {{\mathbb {R}}}\). In such a case, variable y will not be bolded. In order to be succinct, we will sometimes forego the use of a 2-tuple and simply use a single variable \(\mathbf {x} \in X \times {{\mathbb {R}}}\). In such a case, the bolded variable \(\mathbf {x}\) reminds that \(\mathbf {x} \in X \times {{\mathbb {R}}}\). Throughout, \(\kappa :X \times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) is a closed gauge in the sense of [5]. In other words \(\kappa \) is convex, and
For an operator \(U:X\times {{\mathbb {R}}}\rightarrow X\times {{\mathbb {R}}}\), \(\mathrm{Fix}U:=\{\mathbf {x} \in X \times {{\mathbb {R}}}\;|\; U(\mathbf {x})=\mathbf {x}\}\) is its fixed point set. For a function \(f:X \rightarrow {{\mathbb {R}}}_+\), \(\mathrm{argmin}f:=\{x \;|\; f(x)=\inf f(X) \}\) is its set of global minimizers, \(\mathrm{dom}f := \{x\;|\; f(x)< \infty \}\) is its domain, \(\mathrm{lev}_{\le r}(f):=\{x \;|\; f(x)\le r \}\) is its r-lower level set, and \(\mathrm{zer}f := \{x \;|\; f(x)= 0 \}\) is its zero set. For a gauge \(\kappa : X \times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}_+\), the definitions of \(\mathrm{argmin}\kappa \), \(\mathrm{dom}\kappa \), \(\mathrm{lev}_{\kappa < r}\kappa \), and \(\mathrm{zer}\kappa \) are respectively analogous subsets of \(X \times {{\mathbb {R}}}\). For a closed, convex subset \(C \subset X \times {{\mathbb {R}}}\), \(\mathrm{cone}C := \{\lambda \mathbf {x} \;|\; (\mathbf {x},\lambda ) \in (C \times [0,\infty [) \}\) is the cone of C, \(P_C:\mathbf {x} \mapsto \mathrm{argmin}_{\mathbf {y} \in C}\Vert \mathbf {y}-\mathbf {x}\Vert \) is the projection operator associated with C, and \(N_C(\mathbf {x})\) is the normal cone to C at a point \(\mathbf {x} \in C\). For projection operators associated with (closed, convex) lower level sets, we use the shorthand: \(P_{f \le r}:=P_{\mathrm{lev}_{\le r}(f)}\).
We will make use of various notions of nonexpansivity, which we now introduce; more information may be found in [2], and a comparison of what may be shown through different cutter and projection methods is found in [4]. The following definition may be found in either of these.
Definition 1
(Properties of operators). Let \(D\subset X\times {{\mathbb {R}}}\) be nonempty and let \(U:D\rightarrow X\times {{\mathbb {R}}}\). Assume that \(\mathrm{Fix}U \ne \emptyset \). Then U is said to be
-
1.
firmly nonexpansive if
$$\begin{aligned} \Vert U(\mathbf {x})-U(\mathbf {y})\Vert ^2 + \Vert (\mathrm{Id}-U)(\mathbf {x})-(\mathrm{Id}-U)(\mathbf {y})\Vert ^2 \le \Vert \mathbf {x}-\mathbf {y}\Vert ^2 \;\; \forall \mathbf {x} \in D, \;\; \forall \mathbf {y} \in D; \end{aligned}$$ -
2.
nonexpansive if it is Lipschitz continuous with constant 1,
$$\begin{aligned} \Vert U(\mathbf {x})-U(\mathbf {y})\Vert \le \Vert \mathbf {x}-\mathbf {y}\Vert \qquad \forall \mathbf {x} \in D, \quad \forall \mathbf {y} \in D; \end{aligned}$$ -
3.
quasinonexpansive (QNE) if
$$\begin{aligned} \qquad \Vert U(\mathbf {x})-\mathbf {y}\Vert \le \Vert \mathbf {x}-\mathbf {y}\Vert \qquad \forall \mathbf {x} \in D, \quad \forall \mathbf {y} \in \mathrm{Fix}U \end{aligned}$$(an operator that is both quasinonexpansive and continuous is called paracontracting);
-
4.
firmly quasinonexpansive (FQNE) (or a cutter) if
$$\begin{aligned} \Vert U\mathbf {x}-\mathbf {y}\Vert ^2 + \Vert U\mathbf {x}-\mathbf {x}\Vert ^2 \le \Vert \mathbf {x}-\mathbf {y}\Vert ^2\quad \forall \mathbf {x} \in D, \quad \forall \mathbf {y} \in \mathrm{Fix}U; \end{aligned}$$ -
5.
strictly quasinonexpansive (SQNE) if
$$\begin{aligned} \Vert U(\mathbf {x})-\mathbf {y}\Vert < \Vert \mathbf {x}-\mathbf {y}\Vert \qquad \forall \mathbf {x} \in D \setminus \mathrm{Fix}U,\quad \forall \mathbf {y} \in \mathrm{Fix}U; \end{aligned}$$ -
6.
\(\rho \)-strongly quasinonexpansive for \(\rho > 0\) if
$$\begin{aligned} \Vert U\mathbf {x}-\mathbf {y}\Vert ^2 \le \Vert \mathbf {x}-\mathbf {y}\Vert ^2 - \rho \Vert U\mathbf {x} -\mathbf {x}\Vert ^2 \qquad \forall \mathbf {x} \in D \setminus \mathrm{Fix}U,\quad \forall \mathbf {y} \in \mathrm{Fix}U. \end{aligned}$$
Lemma 2.1
[2, Proposition 4.4] Let \(D \subset X \times {{\mathbb {R}}}\) be nonempty. Let \(U:D \rightarrow X\times {{\mathbb {R}}}\). The following are equivalent:
-
1.
U is firmly quasinonexpansive;
-
2.
\(2U-\mathrm{Id}\) is quasinonexpansive;
-
3.
\((\forall \mathbf {x} \in D)(\forall \mathbf {y} \in \mathrm{Fix}U)\quad \Vert U\mathbf {x}-\mathbf {y}\Vert ^2 \le \langle \mathbf {x}-\mathbf {y}, U\mathbf {x}-\mathbf {y}\rangle \);
-
4.
\((\forall \mathbf {x} \in D)(\forall \mathbf {y} \in \mathrm{Fix}U)\quad \langle \mathbf {y}-U\mathbf {x}, \mathbf {x}-U\mathbf {x}\rangle \le 0\);
-
5.
\((\forall \mathbf {x} \in D)(\forall \mathbf {y} \in \mathrm{Fix}U)\quad \Vert U\mathbf {x}-\mathbf {x}\Vert ^2 \le \langle \mathbf {y}-\mathbf {x},U\mathbf {x}-\mathbf {x}\rangle \).
Definition 2
(Fejér monotonicity [2, 5.1]). A sequence \((\mathbf {x}_n)_{n\in {{\mathbb {N}}}}\) is Fejér monotone with respect to a closed convex set \(C \subset X \times {{\mathbb {R}}}\) if
A Fejér monotone sequence with respect to a closed convex set C may be thought of as a sequence defined by \(\mathbf {x}_n := U^n \mathbf {x}_0\) where U is QNE with respect to \(C=\mathrm{Fix}U\). We will make use of the fact that a Fejér monotone sequence with respect to a non-empty set is always bounded. We will also make use of the following convergence result.
Theorem 2.2
[2, Theorem 5.11] Let \((\mathbf {x}_n)_{n\in \mathbb {N}}\) be a sequence in \(X\times {{\mathbb {R}}}\) and let C be a nonempty closed convex subset of \(X\times {{\mathbb {R}}}\). Suppose that \((\mathbf {x}_n)_{n\in \mathbb {N}}\) is Fejér monotone with respect to C. Then the following are equivalent:
-
1.
the sequence \((\mathbf {x}_n)_{n\in \mathbb {N}}\) converges strongly (i.e. in norm) to a point in C;
-
2.
\((\mathbf {x}_n)_{n\in \mathbb {N}}\) possesses a strong sequential cluster point in C;
-
3.
\(\underset{n\rightarrow \infty }{\mathrm{\liminf }}\, d(\mathbf {x}_{n},C)=0.\)
We will make use of the following result, which may be recognized as a simplified version of [4, Theorem 4] and variants of which may be found in [3].
Lemma 2.3
Let \(U: X\times R \rightarrow X \times {{\mathbb {R}}}\) be FQNE. Then the operator given by
is QNE for all \(\gamma \in \left[ 0,2\right] \). Moreover, if \(\gamma \in \left]0,2 \right[\) then U is SQNE and the sequence \((\mathbf {x}_n)_{n \in {{\mathbb {N}}}}\) given by \(\mathbf {x}_n := U^n \mathbf {x}_0\) satisfies
3 The polar envelope and proximity operator
Friedlander, Macêdo, and Pong introduced the polar envelope and its associated polar proximity operator [6], which we now recall.
Definition 3
(Polar envelope and polar proximal map [6]). For any closed gauge \(\kappa : X\times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) and positive scalar \(\alpha \), the function
is the polar envelope of \(\kappa \). The corresponding polar proximal map
sends a point \(\mathbf {x}\) to the minimizing set that defines \(\kappa _\alpha (\mathbf {x})\). Naturally,
denotes the polar envelope of the closed perspective transform \(f^\pi \) for a proper convex function \(f:X \rightarrow {{\mathbb {R}}}\).
Figure 1 shows the construction of the polar envelope and its proximity operator for \(\kappa =\Vert \cdot \Vert _\infty \). At top and at bottom left, we take three choices of \(\mathbf {x}\) and plot the functions \(\Vert \cdot - \mathbf {x}\Vert \) in yellow, red, and orange respectively. The domain points for which each of these epigraphs intersects the epigraph of \(\Vert \cdot \Vert _\infty \) at lowest height are the respective proximal points. The height at the point of intersection determines the envelope value. For points \(\mathbf {x}\) in the white regions, such as the points for which the functions \(\Vert \cdot -\mathbf {x}\Vert \) are orange and yellow respectively, the envelope value is simply \(\Vert \mathbf {x}\Vert _\infty /2\). For points lying in the red regions, the proximal point lies on the diagonals; for points in the interiors of the red regions, the envelope values are strictly greater than \(\Vert \mathbf {x}\Vert _\infty /2\). This results in the smoothing apparent in the red regions for the envelope shown at bottom right.
Construction of the polar envelope and corresponding polar proximal map for \(\kappa =\Vert \cdot \Vert _\infty \)
We devote the remainder of this section to showing that the polar proximity operator \(T_{\kappa ,\alpha }\) is firmly quasinonexpansive. We have the symmetry of vertical rescaling,
by which the proximity operators satisfy \(T_{\kappa ,\alpha } = T_{\alpha \kappa ,1}\), while the envelopes satisfy \((\alpha \kappa )_1 = \alpha (\kappa _\alpha )\). Thus, by working with a general \(\kappa \), we can, and do, let \(\alpha =1\) without loss of generality. For simplicity, we also write T instead of \(T_{\kappa ,\alpha }\). Note that we still need the notation \(\kappa _\alpha \) to distinguish the polar envelope from the gauge \(\kappa \) itself.
Lemma 3.1
For a closed gauge \(\kappa \), it holds that
Proof
The fact that \(\mathrm{zer}\kappa \subset \mathrm{Fix}T\) is obvious. We will show the reverse inclusion. Let \(\mathbf {x} \in \mathrm{Fix}T\). Then
and so
Thus \(\kappa (\mathbf {x}) \le 0\). Combining with the fact \(\kappa (\mathbf {x})\ge 0\), we have \(\kappa (\mathbf {x}) = 0\). \(\square \)
Lemma 3.2
Let \(\kappa \) be a closed gauge and \(\kappa _\alpha \) its polar envelope. Then
Proof
Let \(\kappa (\mathbf {x}) = 0\). Then
Thus \(\kappa _\alpha (\mathbf {x}) = 0\).
Now let \(\kappa _\alpha (\mathbf {x}) = 0\). Since \(\kappa _\alpha (\mathbf {x}) = 0\), there exists a sequence \((\mathbf {u}_n)_{n \in N}\) such that
Then we have that \(\kappa (\mathbf {u}_n) \rightarrow 0\) and \(\Vert \mathbf {x}-\mathbf {u}_n\Vert \rightarrow 0\). Since \(\Vert \mathbf {x}-\mathbf {u}_n\Vert \rightarrow 0\), we have that \(\mathbf {u}_n \rightarrow x\). Combining with the fact that \(\kappa \) is lower semicontinuous, we have that \(\kappa (\mathbf {x}) \le \lim _{\mathbf {u}_n \rightarrow \mathbf {x}} \kappa (\mathbf {u}_n) = 0\). This concludes the result. \(\square \)
Lemma 3.3
Let \(\kappa \) be a closed gauge and \(\kappa (\mathbf {x}) > 0\). Then one of the following holds:
-
(i)
\(\kappa (T\mathbf {x}) <\frac{1}{\alpha }\Vert T\mathbf {x}-\mathbf {x}\Vert \), in which case \(0 \in \frac{1}{\alpha } \frac{T\mathbf {x}-\mathbf {x}}{\Vert T\mathbf {x}-\mathbf {x}\Vert } + N_{\mathrm{dom}\kappa }(T\mathbf {x})\) and \(T\mathbf {x} = P_{\mathrm{dom}\kappa }\mathbf {x}\).
-
(ii)
We have that
$$\begin{aligned} r :=&\kappa (T\mathbf {x}) = \frac{1}{\alpha }\Vert T\mathbf {x}-\mathbf {x}\Vert \\ \text {where}\quad T\mathbf {x} =&P_{\mathrm{lev}_{\le r}(\kappa )}\mathbf {x}, \end{aligned}$$and there exists \(\lambda \in [0,1[\) such that
$$\begin{aligned} 0&\in \frac{1-\lambda }{\alpha } \frac{T\mathbf {x}-\mathbf {x}}{\Vert T\mathbf {x}-\mathbf {x}\Vert } + \partial (\lambda ^+ \kappa )(T\mathbf {x}) \nonumber \\&= \frac{1-\lambda }{\alpha } \frac{T\mathbf {x}-\mathbf {x}}{\Vert T\mathbf {x}-\mathbf {x}\Vert } + {\left\{ \begin{array}{ll} \left\{ \lambda z \; | \; z \in \partial \kappa (T\mathbf {x}) \right\} &{} \text {if}\; \lambda > 0 \\ N_{\mathrm{dom}\kappa }(T\mathbf {x}) &{} \text {if}\; \lambda = 0 \end{array}\right. }. \end{aligned}$$
Proof
Let \(\kappa (\mathbf {x})>0\). From Lemma 3.2, \(\kappa _\alpha (\mathbf {x})>0\). We have from [6, Section 5] that the condition \(\kappa _\alpha (\mathbf {x})>0\) guarantees that one of (i) or (ii) must hold. \(\square \)
We will use the characterization of T from Lemma 3.3 often; hence the shorthand \(P_{\kappa \le r}:=P_{\mathrm{lev}_{\le r}(\kappa )}\). Now we have the principle result of this section, which establishes that the polar proximity operator T is firmly quasinonexpansive.
Theorem 3.4
T is firmly quasinonexpansive.
Proof
Let \(\mathbf {y} \in X \times {{\mathbb {R}}}\) and \(\mathbf {x} \in \mathrm{Fix}T\). If \(\kappa (\mathbf {y}) = 0\) then \(\mathbf {y} \in \mathrm{Fix}T\), so let \(\kappa (\mathbf {y}) > 0\). By Lemma 3.3, we need only consider two cases.
Case 1 If Lemma 3.3(i) holds, then we have that
and so \(\mathbf {y} - T\mathbf {y} \in N_{\mathrm{dom}\kappa }(T\mathbf {y})\). Thus \(T\mathbf {y} = P_{{\overline{\mathrm{dom}\kappa }}}(\mathbf {y})\). The operator \(P_{{\overline{\mathrm{dom}\kappa }}}\) is FQNE with \(\mathrm{Fix}P_{{\overline{\mathrm{dom}\kappa }}} = {\overline{\mathrm{dom}\kappa }}\). Thus by Lemma 2.1,
Since \(\mathbf {x} \in \mathrm{Fix}T \subset {\overline{\mathrm{dom}\kappa }}= \mathrm{Fix}P_{{\overline{\mathrm{dom}\kappa }}}\), we have that (3.2) is true, in particular, for \(\mathbf {u}=\mathbf {x}\) and \(\mathbf {v}=\mathbf {y}\). Thus we obtain
This is just
By Lemma 2.1, this is what we needed to show.
Case 2 If Lemma 3.3(ii) holds with \(\lambda = 0\), then we again obtain (3.1) and proceed as in Case 1, obtaining what we needed to show. If Lemma 3.3(ii) holds with \(\lambda > 0\) then there exists \(r >0\) such that
Now since \(P_{\kappa \le r}\) is FQNE, we have that
Since \(\mathbf {x} \in \mathrm{Fix}T =\mathrm{lev}_{\le 0}\kappa \subset \mathrm{lev}_{\le r_2}\kappa = \mathrm{Fix}P_{\kappa \le r_2}\), we have that (3.3) is true, in particular, for \(\mathbf {u}=\mathbf {x}\) and \(\mathbf {v}=\mathbf {y}\). Thus we obtain
This is just
By Lemma 2.1, this is what we needed to show. \(\square \)
4 The projected polar proximal point algorithm
We now recall the projected polar proximal point algorithm.
Definition 4
(Projected polar proximal point algorithm \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) [6, 7.2]). Fix \(\alpha >0\) and set
The projected polar proximal point algorithm is to begin with any \(v_0\) and update by
Intuitively, the motivation of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) is to minimize a function f by attacking the gauge given by its closed perspective transform \(f^\pi \). Its polar envelope \(f_\alpha ^\pi \) serves a role analogous to the role played by the Fenchel–Moreau envelope in the construction of the traditional proximal point algorithm; see the remarks in [6, 7.2]. The use of the gauge allows problems to be reformulated using gauge duality [5].
In addition to explaining these connections, Friedlander, Macêdo, and Pong also showed that \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) has a useful fixed point property, which we now recall.
Theorem 4.1
(Fixed points of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) [6, Theorem 7.4]). Let \(f:X \rightarrow {{\mathbb {R}}}_+ \cup \{+\infty \}\) be a proper closed nonnegative convex function with \(\inf f>0\) and \(\mathrm{argmin}f \ne \emptyset \). The following hold
-
(i)
If \((v,\lambda _*)=\mathfrak {P}_{1,f}\), then \(\lambda _* > 0\) and \(\lambda _*^{-1}v \in \mathrm{argmin}f\).
-
(ii)
If \(v \in \mathrm{argmin}f\), then there exists \(\lambda _* > 0\) so that \((\tau v, \lambda _*) = \mathfrak {P}_{1,f}(\tau v)\) where \(\tau := \left[ 1+f(v) \right] ^{-1}\).
To show convergence of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\), we will analyse a generalization of it, which we now introduce.
Definition 5
(Generalized projected polar proximal point algorithm (\({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\))). For a gauge \(\kappa : X\times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) and fixed \(\alpha >0\), choose a starting point \(\mathbf {x}_0 \in X\times {{\mathbb {R}}}\) and iterate by
is simply the projection operator for S.
Note that \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) and \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) are defined on X and \(X \times {{\mathbb {R}}}\) respectively. However, when \(\kappa = f^\pi \) the sequences \((v_k)\) from (4.1) and \((\mathbf {x}_k)_k\) from (4.2) clearly satisfy \((v_k,1) = \mathbf {x}_k\), and so the algorithms \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) and \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) generate the same sequence on the non-lifted space. This means that we can study \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) by studying \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\), because their performance for \(\kappa = f^\pi \) is the same.
For our purposes, the advantage of \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) is that it is defined on the lifted space \(X \times {{\mathbb {R}}}\). This allows us to decompose the method into iterative application of the two separate operators: T and \(P_S\). This allows for greater flexibility. Specifically, it allows us to study \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) as a splitting method, where two different operators are applied in succession: first the one and then the other. This allows us to build new characterizations of fixed points, and these characterizations, in turn, allow us to show global convergence. Thus, by analysing \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\), we are able to prove the desired convergence of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) without assuming strong convexity of \((f^\pi )^2\). Whether the added flexibility of \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) has benefits beyond its utility for learning about \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) is a natural question for future research. For the present, our main motivation for introducing \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) is the aforementioned advantage.
4.1 Alternative fixed point characterization
We next establish a useful characterization of the fixed points (Proposition 4.4). For the purpose, we need the following two lemmas.
Lemma 4.2
Let \((x,1) \in \mathrm{Fix}P_S \circ T\). Then the following hold.
-
(i)
\(T(x,1) = (x,\lambda )\) for some \(\lambda \in \left[ 0,1 \right] \).
-
(ii)
Moreover, if \((x,1) \notin \mathrm{Fix}T\), then \(\lambda \in [0,1[\).
Proof
(i): Since \((x,1) \in \mathrm{Fix}P_S \circ T\), we have that
Since \(P_S^{-1}(w,1) = \{(w,\lambda )\; | \lambda \in {{\mathbb {R}}}\}\) for all \(w \in X\), we have that \(T(x,1) = (x,\lambda )\) for some \(\lambda \in {{\mathbb {R}}}\). We now show that \(\lambda \in \left[ 0,1\right] \). Since T is FQNE,
In particular, \(\mathbf {0} \in \mathrm{Fix}T\) and so (4.3) yields
In particular (4.4) holds for \(\mathbf {u}=(x,1)\), and so we have
This is just
Thus \(\lambda \in \left[ 0,1 \right] \).
(ii): Since T is FQNE, it is SQNE [2], and so we have that
Specifically, (4.5) holds for \(\mathbf {u} = (x,1)\) and \(\mathbf {v}=\mathbf {0}\), and so we obtain
This is just
and so we conclude that \(\lambda < 1\). This concludes the result. \(\square \)
Lemma 4.3
Let \((x,1) \in \mathrm{Fix}P_S \circ T\) and \((x,\lambda ) := T(x,1)\) (a representation that always holds by Lemma 4.2). Then for any \((y,1) \in S\) the following hold:
-
(i)
\(\Vert T(y,1)-(y,1)\Vert \ge \Vert T(x,1)-(x,1) \Vert \);
-
(ii)
Additionally, if \(T(x,1) = P_{{\overline{\mathrm{dom}\kappa }}}(x,1)\) or \(\lambda =1\) then
-
(a)
\(\Vert T(y,1)-P_S\circ T(y,1)\Vert \ge \Vert T(x,1)-(x,1)\Vert \);
-
(b)
\(T(y,1) = (w,\mu )\) for some \(\mu \in {{\mathbb {R}}}\) satisfying \(|1-\lambda | \le |1-\mu |\);
-
(c)
If \(\lambda < 1\), then \(T(y,1) = (w,\mu )\) for some \(\mu \le \lambda \).
-
(a)
Proof
Let \((y,1) \in S\) and \((w,\mu ) = T(y,1)\). Set \(r':=\Vert T(x,1)-(x,1) \Vert \) and \(r_2:=\Vert T(y,1)-(y,1)\Vert \). We have by Lemma 4.2 that
We will consider two cases: when \(\lambda = 1\) and when \(\lambda \in [0,1[\).
Case \(\lambda =1\) Then \(|1-\lambda | = 0\), and so (ii)b clearly holds. Additionally, by Lemma 4.2, we have that \((x,1) \in \mathrm{Fix}T\), and so \(\Vert T(x,1)-(x,1)\Vert = 0\), and so (i) and (ii)a both clearly hold, while (ii)c does not apply. This concludes what we needed to show in the case \(\lambda =1\).
Case \(\lambda <1\) Let
By Lemma 3.3, we may further consider two subcases, namely when \(T(x,1) = P_{{\overline{\mathrm{dom}\kappa }}}(x,1)\) and when \(T(x,1) = P_{\kappa \le r'}(x,1)\).
Subcase 1 Let \(T(x,1) = P_{{\overline{\mathrm{dom}\kappa }}}(x,1)\). Since \(P_{{\overline{\mathrm{dom}\kappa }}}\) is FQNE with \(\mathrm{Fix}P_{{\overline{\mathrm{dom}\kappa }}} = {\overline{\mathrm{dom}\kappa }}\), we have that
In particular, (4.7) holds for \(\mathbf {u} = (x,1)\) and \(\mathbf {v} = T(y,1)\). Thus
Since \(P_{{\overline{\mathrm{dom}\kappa }}}(x,1) = T(x,1)\), we have \(P_{{\overline{\mathrm{dom}\kappa }}}(x,1) = (x,\lambda )\). Combining with (4.8), we obtain
Recall that by (4.6), we have \(\lambda \in [0 ,1[\). Thus we have \((\lambda -1) < 0\). Combining this fact with (4.10), we have \(\lambda - \mu \ge 0\), and so \(\lambda \ge \mu \). This shows (ii)b and (ii)c. Thus \((1-\mu ) \ge (1-\lambda ) > 0\), where the final inequality is because \(\lambda < 1\). Thus we have
We also have that
and that
Combining (4.11), (4.12), and (4.13), we obtain \(\Vert T(y,1) - (y,1)\Vert \ge r'\). Thus \(r_2 \ge r'\), which shows (i).
We also have that
which shows (ii)a. Thus we have shown everything we needed to show in the case when \(T(x,1) = P_{{\overline{\mathrm{dom}\kappa }}}(x,1)\)
Subcase 2 Let \(T(x,1) = P_{\kappa \le r'}(x,1)\). Suppose for a contradiction that \(r_2 < r'\). For the sake of simplicity, define \(P_{r'}:=P_{\kappa \le r'}\). We have that
where the first inequality is because \(\kappa (T(y,1)) \le \Vert T(y,1)-(y,1)\Vert \) [6, Theorem 4.4] and the second is our contradiction assumption. Thus
As \(\mathrm{lev}_{\le r'}\kappa \) is closed and convex, the operator \(P_{r'}\) is FQNE with \(\mathrm{Fix}P_{r'} = \mathrm{lev}_{\le r'}\kappa \). Using Lemma 2.1, we have that
In particular, \(T(y,1) \in \mathrm{Fix}P_{\kappa \le r'}\), and so we can apply (4.14) with \(\mathbf {u} = (x,1)\) and \(\mathbf {v} = T(y,1)\), obtaining
Since \(P_{r'}(x,1) = T(x,1)\), we have \(P_{r'}(x,1) = (x,\lambda )\). Let \((w,\mu ):= T(y,1)\). Combining with (4.15), we again obtain (4.9), and we proceed as we did in Case 1 to obtain \(r_2 \ge r'\), a contradiction. Thus \(r_2 \ge r'\), which shows (i). \(\square \)
We now establish the useful alternative characterization of the set \(\mathrm{Fix}P_S \circ T\).
Proposition 4.4
(Fixed points of \(P_S\circ T\)). Let \((x,1) \in \mathrm{Fix}P_S \circ T\) and set \(r' := \Vert (x,1)-T(x,1)\Vert \). It holds that
Proof
The first inclusion \(\{(u,1) \; | \; \Vert T(u,1)-(u,1)\Vert = r' \} \supset \mathrm{Fix}P_S \circ T\) is a consequence of Lemma 4.3. Simply let \((x,1),(y,1) \in \mathrm{Fix}P_S \circ T\), and we have from Lemma 4.3(i) that
and so \(\Vert T(y,1)-(y,1)\Vert = \Vert T(x,1)-(x,1)\Vert = r'\).
Now we will show the reverse inclusion. Let \((y,1) \in S\) and let
By Lemma 4.2 we have that
Case 1 \(\lambda =1\). Suppose \(\lambda =1\). Then we have that \((x,1) \in \mathrm{Fix}T\) and so \(r' = 0\). Since \(\Vert T(y,1)-(y,1)\Vert =r'\), we have \(\Vert T(y,1)-(y,1)\Vert = 0\). Thus \((y,1) \in \mathrm{Fix}T\), and so \((y,1) \in \mathrm{Fix}P_S \circ T\). This concludes the case when \(\lambda =1\).
Case 2 \(\lambda \in [0,1[\). Let \(\lambda \in [0,1[\). Then \((x,1) \notin \mathrm{Fix}T\) and so \(\kappa (x,1) > 0\). Since \(\kappa (x,1)>0\), we have by Lemma 3.3 that either \(T(x,1) = P_{{\overline{\mathrm{dom}\kappa }}}(x,1)\) or \(T(x,1) = P_{\kappa \le r'}(x,1)\).
Case 2(a) \(T(x,1) = P_{{\overline{\mathrm{dom}\kappa }}}(x)\). Since \({\overline{\mathrm{dom}\kappa }}\) is closed and convex, we have that \(P_{{\overline{\mathrm{dom}\kappa }}}\) is FQNE. Since \(P_{{\overline{\mathrm{dom}\kappa }}}\) is FQNE with \(\mathrm{Fix}P_{{\overline{\mathrm{dom}\kappa }}} = {\overline{\mathrm{dom}\kappa }}\), we have that
In particular, \(T(y,1) \in {\overline{\mathrm{dom}\kappa }}\), and so we may apply (4.17) holds with \(\mathbf {u} = T(y,1)\) and \(\mathbf {v} = (x,1)\), obtaining
Using the fact that \(T(y,1) = (w,\mu )\) and \(P_{{\overline{\mathrm{dom}\kappa }}}(x,1) = T(x,1) = (x,\lambda )\), (4.18) becomes
From (4.19) we have that
Using the fact that \(\lambda < 1\), (4.20) implies that \(\mu \le \lambda \). Thus we have that \(1-\mu \ge 1-\lambda \ge 0\). Thus we have that
Now since
we have that
Combining (4.21) and (4.22), we obtain
Thus we have that \(w=y\), and so \(T(y,1) = (y,\mu )\). Thus
and so \((y,1) \in \mathrm{Fix}P_S \circ T\).
Case 2(b) \(T(x,1) = P_{\kappa \le r'}(x,1)\). Let \(T(x,1) = P_{\kappa \le r'}(x,1)\).
Since \(P_{\kappa \le r'}\) is FQNE with \(\mathrm{Fix}P_{\kappa \le r'} = \mathrm{lev}_{\le r'}\kappa \), we have that
In particular, since \(\kappa (T(y,1))\le \Vert T(y,1)-(y,1)\Vert =r'\), we have that \(T(y,1) \in \mathrm{Fix}P_{\kappa \le r'}\), and so we may apply (4.23) with \(\mathbf {u} = T(y,1)\) and \(\mathbf {v} = (x,1)\), obtaining
Using the fact that \(T(y,1) = (w,\mu )\) and \(P_{\kappa \le r'}(x,1) = T(x,1) = (x,\lambda )\), (4.24) again yields (4.19), and we proceed as in Case 2(a).
This shows the desired result. \(\square \)
The following shorthand will simplify notation in the results that follow. Whenever \(\mathrm{Fix}P_S \circ T \ne \emptyset \), we define
Our previous results admit the following important property.
Lemma 4.5
Whenever \(\mathrm{Fix}P_S \circ T \ne \emptyset \), we have that
and equality holds if and only if \(\mathbf {y} \in \mathrm{Fix}P_S \circ T\).
Proof
The inequality is an immediate consequence of Lemma 4.3(i). The fact that equality holds if and only if \(\mathbf {y} \in \mathrm{Fix}P_S \circ T\) is an immediate consequence of Proposition 4.4. \(\square \)
4.2 Convergence
In this subsection, we will show that sequences admitted by \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) are globally convergent to a point in \(\mathrm{Fix}P_S \circ T\), whenever the latter is nonempty. The key result, Theorem 4.8, uses the following auxiliary lemma.
Lemma 4.6
Let \(\mathrm{Fix}P_S \circ T \ne \emptyset \). The following holds:
Proof
Let \(\mathbf {x} \in \mathrm{Fix}P_S \circ T\) and \(\mathbf {y} \in S\). By Lemma 3.3, we may consider two cases: when \(T\mathbf {y} = P_{{\overline{\mathrm{dom}\kappa }}}\mathbf {y}\) and when \(T\mathbf {y} = P_{\kappa \le r_2}\mathbf {y}\) where \(r_2 = \Vert T\mathbf {y}-\mathbf {y}\Vert \).
Case 1 Let \(T\mathbf {y} = P_{{\overline{\mathrm{dom}\kappa }}}\mathbf {y}\). Since \(P_{{\overline{\mathrm{dom}\kappa }}}\) is FQNE, we have from Lemma 2.1 that
In particular \(T\mathbf {x} \in \mathrm{Fix}P_{{\overline{\mathrm{dom}\kappa }}}\), and so we can apply (4.26) with \(\mathbf {u} = T\mathbf {x}\) and \(\mathbf {v} = \mathbf {y}\), obtaining
Finally, substituting in (4.27) using the fact that \(P_{{\overline{\mathrm{dom}\kappa }}}\mathbf {y} = T\mathbf {y}\), we obtain
This shows the result in Case 1.
Case 2 Let \(T\mathbf {y} = P_{\kappa \le r_2}\mathbf {y}\) where \(r_2 = \Vert T\mathbf {y}-\mathbf {y}\Vert \). As \(P_{\kappa \le r_2}\) is FQNE, we have that
We have by Lemma 4.3 that \(r_2 \ge r' = \Vert T\mathbf {x}-\mathbf {x}\Vert \). Combining this with the fact from [6, Theorem 4.4(ii)] that \(\kappa (T\mathbf {x}) \le \Vert T\mathbf {x}-\mathbf {x}\Vert \) , we have that \(T\mathbf {x} \in \mathrm{lev}_{\le r'} \kappa \subset \mathrm{lev}_{\le r_2}\kappa \). Thus \(T\mathbf {x} \in \mathrm{Fix}P_{\kappa \le r_2}\), and so we can apply (4.28) with \(\mathbf {u}=T\mathbf {x}\) and \(\mathbf {v}=\mathbf {y}\), obtaining
Finally, since \(P_{\kappa \le r_2}\mathbf {y} = T\mathbf {y}\), we may substitute in (4.29) to obtain
This concludes the result. \(\square \)
Fact 4.7
Since S is an affine subspace, it holds that
Proof
This follows immediately from the Pythagorean theorem. \(\square \)
The strategy for Theorem 4.8 is illustrated
Figure 2 illustrates the strategy of the following theorem, which brings together all of the different results we have established so far.
Theorem 4.8
Let \(\mathrm{Fix}P_S \circ T \ne \emptyset \). Set \(\mathbf {y}_0 \in S\). The operator \(P_S \circ T: S \rightarrow S\) is SQNE, and so the sequence \((\mathbf {y}_n)_{n \in {{\mathbb {N}}}} \subset S\) given by
is Fejér monotone with respect to \(\mathrm{Fix}(P_{S} \circ T)\). More specifically,
where E is as in (4.1).
Proof
Let \(\mathbf {x} \in \mathrm{Fix}P_S \circ T\) and \(\mathbf {y} \in S\). There exists \(\beta \in {{\mathbb {R}}}\) such that
We will first show that \(\beta \ge 0\). From Lemma 4.6 we have that
Adding \(\langle \mathbf {y}-T\mathbf {x},\mathbf {y}-T\mathbf {y} \rangle \) to both sides of (4.32) yields
Combining (4.31) and (4.33), we obtain
Now (4.34) implies that \(\beta \ge 0\) or \(\Vert T\mathbf {y}-\mathbf {y}\Vert = 0\). If \(\Vert T\mathbf {y}-\mathbf {y}\Vert = 0\), then \(T\mathbf {y}=\mathbf {y} \in S\) and so \(P_S\circ T \mathbf {y} = \mathbf {y}\), and so \(\mathbf {y} \in \mathrm{Fix}(P_S \circ T)\), in which case we are done. Thus we may restrict to considering the case when \(\Vert T\mathbf {y}-\mathbf {y}\Vert > 0\). This, together with (4.34), yields
Now applying the Pythagorean Theorem to (4.31) yields
Moreover, we may rearrange (4.31) to obtain
Applying the Pythagorean Theorem to (4.36), we obtain
Using (4.35) to substitute for \(\Vert \mathbf {u}\Vert ^2\) in (4.37), we obtain
Now from Fact 4.7 we have that (4.30) holds for \(\mathbf {u}=T\mathbf {x}\) and \(v=\mathbf {y}\), and so we have
Now we may use (4.39) to substitute for \(\Vert T\mathbf {x}-\mathbf {y}\Vert ^2\) in (4.38) and obtain
Now we have that
where E is as defined in 4.1. Multiplying both sides of (4.41) by \(-(1+2\beta )\), we obtain
Using (4.42) to make the appropriate substitution for \(-(1+2\beta )\Vert T\mathbf {y}-\mathbf {y}\Vert ^2\) in (4.40), we obtain
Since S is closed and convex, \(P_S\) is nonexpansive ( [2, Proposition 4.16]). Thus we have that
Since \(\mathbf {x} \in \mathrm{Fix}P_S \circ T\), we have that \(P_S \circ T \mathbf {x} = \mathbf {x}\). Making this substitution in (4.44), we obtain
Together, (4.43) and (4.45) yield
where the second inequality uses the fact that \(\beta \ge 0\) and \(E(\mathbf {x},\mathbf {y}) \ge 0\). This shows the desired result. \(\square \)
Theorem 4.8 admits the following corollary.
Corollary 4.9
(Averaged variant). Let \(\mathrm{Fix}P_S \circ T \ne \emptyset \). The operator given by
is FQNE.
Proof
By Theorem (4.8), we have that \(P_S \circ T\) is QNE. By Lemma 2.1, an operator R is FQNE if and only if \(2U-\mathrm{Id}\) is QNE. Letting
we have that U is FQNE and that \(U=\frac{1}{2} P_S \circ T + \frac{1}{2} \mathrm{Id}\). \(\square \)
Now having the key results of Theorem 4.8, we are ready to show convergence for both \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\) and its under-relaxed variants.
Theorem 4.10
(Convergence of under-relaxed variants of \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\)). Let \(\mathrm{Fix}P_S \circ T \ne \emptyset \). Let \(\gamma \in \left]0,1 \right[\) and \(\mathbf {y}_0 \in S\). The sequence given by
is strongly convergent to some \(\mathbf {y} \in \mathrm{Fix}P_S \circ T\).
Proof
Notice that
where \(2\gamma \in \left]0,2\right[\) and
is the FQNE operator from Corollary 4.9. Thus, applying Lemma 2.3 for the operator U, we have that \(\mathcal {U}_\gamma \) is QNE and that
Since \((\mathbf {y}_n)_{n \in {{\mathbb {N}}}}\) is Fejér monotone, it is bounded. Since S is finite dimensional and \((\mathbf {y}_n)_{n \in {{\mathbb {N}}}} \subset S\) is bounded, we may take a convergent subsequence \((\mathbf {y}_j)_{j \in J \subset {{\mathbb {N}}}}\) such that
for some \(\mathbf {y} \in S\). Since T is continuous and \(P_S\) is continuous, \(P_S\circ T\) is continuous. Thus we have
The triangle inequality yields
Taking the limit as \(j \rightarrow \infty \), each of the terms in the right hand side of (4.51) go to zero by (4.49), (4.48), and (4.50) respectively. Thus \(\Vert \mathbf {y}-P_S \circ T(\mathbf {y})\Vert = 0\), and so \(\mathbf {y} = P \circ T(\mathbf {y})\). Thus we have that \(\mathbf {y} \in \mathrm{Fix}P_S \circ T\).
Since \((\mathbf {y}_n)_{n \in {{\mathbb {N}}}}\) is Fejér monotone with respect to \(\mathrm{Fix}P_S \circ T\) and possesses a sequential cluster point \(\mathbf {y} \in \mathrm{Fix}P_S \circ T\), we conclude by Theorem 2.2 that \(\mathbf {y}_n \rightarrow \mathbf {y}\) as \(n \rightarrow \infty \). \(\square \)
Having proven the convergence for the under-relaxed variants of \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\), we now show the convergence of its non-relaxed version.
Theorem 4.11
(Convergence of \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\)). Let \(\mathrm{Fix}P_S \circ T \ne \emptyset \). Let \(\mathbf {y}_0 \in S\). The sequence \((\mathbf {y}_n)_{n \in {{\mathbb {N}}}}\) given by
converges to a point \(\mathbf {y} \in \mathrm{Fix}P_S \circ T\).
Proof
Fix \(\mathbf {x} \in \mathrm{Fix}P_S \circ T\). Applying Theorem 4.8, we have that
and so we have that
Thus we obtain
which shows that
From the definition of E, (4.52) implies that
As \((\mathbf {y}_n)_{n \in {{\mathbb {N}}}}\) is Fejér monotone, it is bounded. Thus we may take a convergent subsequence \((\mathbf {y}_j)_{j \in J \subset {{\mathbb {N}}}}\). Therefore, let
Combining with (4.53), we obtain
where the first equality follows from the continuity of \(T-\mathrm{Id}\) and the second equality is from (4.53). Now since \(\Vert T\mathbf {y}-\mathbf {y}\Vert = \Vert T\mathbf {x}-\mathbf {x}\Vert \) with \(\mathbf {x} \in \mathrm{Fix}P_S \circ T\), we have by Proposition 4.4 that \(\mathbf {y} \in \mathrm{Fix}P_S \circ T\). Since \((\mathbf {y}_n)_{n \in {{\mathbb {N}}}}\) is Fejér monotone with respect to \(\mathrm{Fix}P_S \circ T\) and possesses a sequential cluster point \(\mathbf {y} \in \mathrm{Fix}P_S \circ T\), we conclude by Theorem 2.2 that \(\mathbf {y}_n \rightarrow \mathbf {y}\) as \(n \rightarrow \infty \). \(\square \)
When \(\mathrm{Fix}P_S \circ T \ne \emptyset \), Theorem 4.11 guarantees the convergence of \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\), and Theorem 4.10 does the same for its under-relaxed variants. The next corollary simply formalizes this by including both cases.
Corollary 4.12
Let \(\mathrm{Fix}P_S \circ T \ne \emptyset \). Let \(\gamma \in [0,1[\) and \(\mathbf {y}_0 \in S\). Then the sequence given by
is convergent to some \(\mathbf {y} \in \mathrm{Fix}P_S \circ T\), and the operator \(\mathcal {U}_\gamma \) is paracontracting.
Proof
The convergence when \(\gamma \in \left]0,1\right[\) is shown by Theorem 4.10, and the convergence when \(\gamma = 0\) is dealt with by Theorem 4.11. These theorems also show that \(\mathcal {U}_\gamma \) is QNE in these two cases respectively. Combining with the fact that \(\mathcal {U}_\gamma \) is obviously a weighted average of continuous operators, the paracontracting property is clear. \(\square \)
4.3 Shadow sequence behaviour
Having established convergence for the governing sequence, we also have the following result that describes the behaviour of the sequence of shadows of the proximity operator: \(T\circ (\mathcal {U}_\gamma )^{n-1}(y_0,1)\).
Corollary 4.13
Let \(\mathrm{Fix}P_S \circ T \ne \emptyset \). Set \((y_0,1) \in S\) and \(\gamma \in [0,1[\). The sequence \((\lambda _n)_{n \in {{\mathbb {N}}}} \subset S\) given by
where \(\mathcal {U}_\gamma \) is as specified in (4.54) satisfies
where \(r'\) is as specified in (4.16).
Proof
From Corollary 4.12 we have that
Given that \(\mathbf {x} =(y,1)\) for some \(y \in X\), we clearly have
From Proposition 4.4, we have that
where \(r'\) is as characterized in (4.16). Using Lemma 4.2, we have that \(T(y,1)=(y,\lambda )\) for some \(\lambda \in \left[ 0,1\right] \). This yields
Combining (4.55) and (4.56) we have that \(r' = 1-\lambda \), and so
Since \((y_n,1) \rightarrow (y,1)\) and T is continuous, we have that
Thus \(\lambda _n \rightarrow \lambda = 1-r'\). This concludes the result. \(\square \)
4.4 The operator \(P_S \circ T\) is not, generically, FQNE
The property of firm quasinonexpansivity is especially important in the analysis of algorithms. In this section, we discuss under which conditions the operator \(P_S \circ T\) may or may not exhibit this property. In particular, we provide an example illustrating that it is not, generically, FQNE. First we show, in Proposition 4.14, that failure to be a FQNE operator implies some specific conditions.
Proposition 4.14
Let \((y,1) \in S\), and let \((x,1) \in \mathrm{Fix}P_S \circ T\). Let \((x,\lambda ) = T(x,1)\) and \((w,\mu )=T(y,1)\). If
then the following hold:
-
(i)
\(\lambda< \mu < 1\);
-
(ii)
\(T(x,1) \ne P_{{\overline{\mathrm{dom}\kappa }}}(x,1)\).
Proof
Suppose that (4.57) holds. Then we have that
We will first show (i).
From Lemma 4.6 we have that
Let \((w,\mu ) = T(y,1)\) and \((x,\lambda )=T(x,1)\) for \(\lambda \le 1\). Then (4.60) becomes
By Lemma 4.2(i), \(\lambda \in \left[ 0,1\right] \) and so there are only four possibilities: when \(\lambda =1\), when \(\mu \le \lambda <1\), when \(\lambda < 1 \le \mu \), and when \(\lambda< \mu < 1\). We will show that any case other than \(\lambda< \mu < 1\) implies a contradiction.
Case \(\lambda =1\) Suppose \(\lambda =1\). Then \((\lambda -\mu )(1-\mu ) = (1-\mu )^2 \ge 0\).
Combining this fact with (4.61), we obtain
which contradicts (4.59), and so we obtain a contradiction. This concludes the case \(\lambda =1\).
Case \(\mu \le \lambda < 1\) Suppose \(\mu \le \lambda \le 1\). We have that \((\lambda -\mu ) \ge 0\) and \((1-\mu )\ge 0\), and so clearly \((\lambda -\mu )(1-\mu ) \ge 0\). Combining this with (4.61), we again obtain (4.62), which is a contradiction. This concludes the case when \(\mu \le \lambda < 1\).
Case \(\lambda < 1 \le \mu \) Suppose \(\lambda < 1 \le \mu \). Then \((\lambda -\mu ) \le 0\) and \((1-\mu ) \le 0\), and so \((\lambda -\mu )(1-\mu ) \ge 0\). Combining this with (4.61), we again obtain (4.62), a contradiction. This concludes the case when \(\lambda < 1 \le \mu \).
We are left with only one possibility, \(\lambda< \mu < 1\), and so (i) holds.
We next show (ii). Having established that \(\lambda < \mu \), we have by Lemma 4.3(ii)c that \((\lambda < \mu ) \implies T(x,1) \ne P_{{\overline{\mathrm{dom}\kappa }}}(x,1)\). Thus we have (ii). \(\square \)
Proposition 4.14 shows that any example of a gauge for which \(P_S \circ T\) fails to be firmly quasi-nonexpansive must satisfy both (i) and (ii). Now we will see an example that satisfies both of these properties and serves as a counter-example to the tempting idea that \(P_S \circ T\) is generically FQNE.
An illustration of Example 1
Example 1
(\(P_S \circ T\) is generically not a cutter). Let
where \(C:= \{(v,u)\;|\; v\le -u/4 \}\). Then
and so \((1,-1/20) \in \mathrm{Fix}P_S \circ T_\gamma \). Additionally,
and so \((2,1) \notin \mathrm{Fix}P_S \circ T_{\gamma }\). We have that
This example is illustrated in Fig. 3.
5 Fundamental set and existence of fixed points
In the previous section, we consistently assumed that \(\mathrm{Fix}P_S \circ T \ne \emptyset \). It bears noting that this condition may not hold for a general gauge. Of course, it does hold for \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\), under the conditions in Theorem 4.1. We will provide sufficient conditions for the more general \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\), which allow us to describe the solutions to \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) as lying on an exposed face of a dilated fundamental set. For the purpose, we make use of the Minkowski function representation of the gauge:
Such a representation always holds by choosing \(D = \{\mathbf {x} \in X\times {{\mathbb {R}}}\;|\; \kappa (\mathbf {x}) \le 1\}\) [6].
The following Lemma will be instrumental to our main result in Theorem 5.2.
Lemma 5.1
Let \(D= \mathrm{lev}_{\kappa \le 1}\). The following hold.
-
(i)
\(\mathrm{cone}D = \mathrm{dom}\kappa \).
-
(ii)
If there exists \(\lambda ' > 0\) such that
$$\begin{aligned} \lambda ' = \max _{\lambda \in {{\mathbb {R}}}}\{\lambda \;\;|\;\; \exists y' \in X \text {so that}\;(y',\lambda ') \in D \}, \end{aligned}$$then for any \((y,\beta ) \in \mathrm{cone}D \cap (X\times {{\mathbb {R}}}_{>0})\) there exists a minimal \(0<r<\infty \) so that \((y,\beta ) = r \mathbf {d}\) for some \(\mathbf {d} \in D\).
Proof
(i): Let \(\mathbf {x} \in \mathrm{cone}D\). Then there exist \(\lambda <\infty , \mathbf {d} \in D\) such that \(\mathbf {x} = \lambda \mathbf {d}\). By positive homogeneity, \(\kappa (\lambda \mathbf {d})= \lambda \kappa (\mathbf {d}) \le \lambda < \infty \), and so \(\mathbf {x} \in \mathrm{dom}\kappa \), so \(\mathrm{cone}D \subset \mathrm{dom}\kappa \). Now let \(\mathbf {x} \in \mathrm{dom}\kappa \). Then there exists \(r<\infty \) such that \(\kappa (\mathbf {x}) = r\) and so by homogeneity \(\kappa (\mathbf {x}/r) = 1\) and so \(\mathbf {x}/r \in D\), so \(\mathbf {x}= r(\mathbf {x}/r) \in \mathrm{cone}D\), and so \(\mathrm{dom}\kappa \subset \mathrm{cone}D\).
(ii): Let \((y,\beta ) \in \mathrm{cone}D \cap (X\times {{\mathbb {R}}}_{>0})\). First of all, notice that by (i), \((y,\beta ) \in \mathrm{dom}\kappa \) and so
Next we show \(r>0\). Since \((y,\beta ) \in \mathrm{cone}D\), there exists some \(\lambda \ge 0, (d_y,\mu )\in D\) such that \(\lambda (d_y,\mu ) = (y,\beta )\). Now any such \(\lambda \) that satisfies this equality clearly satisfies \(\lambda \mu = \beta \) with \(\beta >0\), and so all three constants are greater than zero. Moreover, any such constant \(\lambda \) that satisfies this equality satisfies \(\lambda = \beta /\mu \ge \beta / \lambda ' > 0\). Thus we have that
Now let \((\lambda _n)_n\) satisfy \(\lambda _n \downarrow r\) as \(n \rightarrow \infty \). Since \(\lambda _n >r\) for all n, \((y,\beta ) \in \lambda _n D\) for all n and so there exists \((\mathbf {d}_n)_n\) such that \((y,\beta )=\lambda _n \mathbf {d}_n\) for all n. Notice that
and so the sequence \((\mathbf {d}_n)_n\) is bounded. Thus we can pass to a convergent subsequence if need be and have \(\mathbf {d}_n \rightarrow \mathbf {d}\) as \(n \rightarrow \infty \). Since D is closed and \(\mathbf {d}_n \in D\) for all n, we have \(\mathbf {d} \in D\). Taking the limit of both sides of
as \(n \rightarrow \infty \), we have \((y,\beta ) = \mathbf {d}r\) with \(\mathbf {d} \in D\) and r being the attained infimum of all such values such that \((y,\beta ) \in rD\). This shows the desired result. \(\square \)
The following theorem provides conditions that guarantee nonemptiness of the fixed point set. The strategy is to relate an exposed face of the fundamental set D to the fixed points of the algorithm.
Theorem 5.2
(Existence of fixed points of \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\)). Let D be the (closed) fundamental set of \(\kappa \) as in (5.1). The following hold.
-
(i)
If there exists \(\lambda ' \ge 0\) such that
$$\begin{aligned} \lambda ' = \max _{\lambda \in {{\mathbb {R}}}}\{\lambda \;\;|\;\; \exists y \in X \text {so that}\;(y,\lambda ) \in D \}, \end{aligned}$$then \(F=D \cap \{(y,\lambda ')\; |\; y \in X\}\) is an exposed face of D and
-
(a)
\(\left( \frac{1}{1+\lambda '} \right) F = T(\mathrm{Fix}P_S \circ T);\) and
-
(b)
Any \((x,1) \in \mathrm{Fix}P_S \circ T\) satisfies \(T(x,1)=\left( x,\frac{\lambda '}{1+\lambda '}\right) \).
For example, this is always the case when D is bounded.
-
(a)
-
(ii)
If such a \(\lambda '\) does not exist and there exists a sequence \((y_n,\lambda _n)_{n \in {{\mathbb {N}}}}\) such that \(\lambda _n\rightarrow \infty \) and \(\lambda _n/\Vert y_n\Vert \rightarrow m>0\), then \(T(\mathrm{Fix}P_S \circ T) = \mathrm{Fix}P_S \circ T = \mathrm{zer}\kappa \cap S\).
Proof
(i) Suppose \(\lambda '\) exists as described. To understand why F is an exposed face of D, see Remark 1 below.
Case 1 \(\lambda '=0\), then any point in F is of the form (x, 0) for some \(x \in X\). Moreover, (0, 1) is a fixed point and satisfies \(\Vert T(0,1)-(0,0)\Vert =1\). It is then a straightforward consequence of Proposition 4.4 that \(T(x,1)=(x,0)\) if and only if (x, 1) is a fixed point of \(P_S\circ T\). This is all we needed to show in this case.
Case 2 \(\lambda '>0\). Since \(\lambda '>0\), the set \(\cup _{r\ge 0}rD\cap (X\times \{\theta \}) \ne \emptyset \) for each \(\theta >0\). For any \((x,\theta ) \in \mathrm{cone}D\) with \(\theta >0\) we have from Lemma 5.1(ii) that there exists minimal \(r_{(x,\theta )}>0\) such that \((x,\theta ) \in r_{(x,\theta )} D.\) By the Minkowski definition of the gauge this means,
There must also exist \((d_x,\mu _\theta ) \in D\) such that \(r_{(x,\theta )}(d_x,\mu ) = (x,\theta )\). Thus \(r_{(x,\theta )}\mu _\theta =\theta \) and \(r_{(x,\theta )} =\theta /\mu _\theta \). Furthermore, our choice of \(\lambda '\) guarantees that \(\mu _\theta \le \lambda '\). Using positive homogeneity we have
Again using positive homogeneity, we obtain
where the final equality is because we just showed \(\kappa (d_x,\mu _\theta )=1\).
Additionally, for any point \((y',\lambda ') \in F\), homogeneity assures that
Now let
Now we show that \((p_{\theta '} y',1)\) is in \(\mathrm{Fix}P_S \circ T\). Remember that \(\kappa _1\) is the polar envelope of \(\kappa \) from Definition 3. We have the following.
Here (5.5a) is true by Lemma 5.1(i), (5.5b) holds by (5.2), (5.5c) holds because \(\mu _{\theta }\le \lambda '\), (5.5d) holds by (5.4), and (5.5e) is obtained by applying (5.3) with \(\theta = \theta '\). Altogether (5.5) shows that \((p_{\theta '}y',\theta ')=T(p_{\theta '}y',1)\), and so \((p_{\theta '}y',1)\in \mathrm{Fix}P_S \circ T\).
Notice that \(\theta ' \in \left]0,1\right[\) and is nearer to 1 for larger \(\lambda '\) and nearer to 0 for smaller \(\lambda '\), exactly as we would expect. Notice also that we have shown that any point
admits a corresponding point \((p_{\theta '}y',1) \in \mathrm{Fix}P_S \circ T\) whose proximal image is
This shows that
Now let \((x,1) \in \mathrm{Fix}P_S \circ T\). Using Proposition 4.4, we have that any \((x,1) \in \mathrm{Fix}P_S \circ T\) must satisfy \(T(x,1)=(x,\lambda )\) where \(\lambda \in \left[ 0,1\right] \) and
which forces \(\lambda = \theta '\). This shows that (i)b is true.
Now by Lemma 5.1(i),\((x,\theta ') \in \mathrm{cone}D\), since \((x,\theta ') \in \mathrm{dom}\kappa \). Now using the fact that \((x,\theta ')\in \mathrm{cone}D\) and following the same reasoning as we used above to obtain (5.2), we have that there exists a value \(r_{(x,\theta ')}\) and a point \((d_x,\mu _{\theta '})\) with \(\mu _{\theta '} \le \lambda '\) such that \(r_{(x,{\theta '})}(d_x,\mu _{\theta '}) = (x,{\theta '})\) and \(\kappa (x,{\theta '})=({\theta '}/\mu _{\theta '})\). Since \((x,1),(p_{\theta '}y',1) \in \mathrm{Fix}P_S \circ T\), we have from Proposition 4.4 that
Moreover, \({\theta '}/\mu _{\theta '}=\kappa (x,{\theta '}) \le \Vert (x,{\theta '})-(x,1)\Vert =r'\), and so
The equality throughout (5.6) forces \(\mu _{\theta '} = \lambda '\). Finally,
This shows that
This concludes the proof of (i)a.
(ii): Let the sequence \((y_n,\lambda _n)_{n \in {{\mathbb {N}}}}\) exist as described. By compactness of the unit ball in Euclidean space and by appealing to a subsequence if necessary, the sequence \(y_n/\Vert y_n\Vert \) converges to some y in the unit ball in X. Now since \((y_n,\lambda _n) \in D\) for all n,
and by the Minkowski function representation of \(\kappa \),
Taking the limits of both sides as \(n\rightarrow \infty \) and using the lower semicontinuity of \(\kappa \), we obtain
The point \(\left( \frac{y}{m},1\right) \in \mathrm{zer}\kappa \cap S\) is clearly a fixed point of T since
Thereafter appealing to Lemma 3.1, Proposition 4.4, and the fact that \(\kappa (T(x,1)) \le \Vert (x,1)-T(x,1)\Vert =r'=0\) for any \((x,1) \in \mathrm{Fix}P_S \circ T\), the result (ii) is clear. \(\square \)
Remark 1
(What do we mean by an exposed face?). Let us explain what we mean in Theorem 5.2 when we say that F is an exposed face of D. F is an exposed face of a closed, convex set D if there exists a supporting hyperplane H to D with \(F = D \cap H\). In our case, \(H = X \times \{\lambda ' \}\). The hyperplane H is a supporting hyperplane because D lies entirely in the affine half space \(X \times {{\mathbb {R}}}_{\le \lambda '}\) defined by H.
The following example showcases a situation when \(\mathrm{Fix}P_S \circ T\) may be empty. In so-doing, it illustrates the importance of the condition \(m>0\) in Theorem 5.2(ii).
Example 2
Let \(D= \{(y,\lambda )\; |\; y \ge \lambda ^2 \} \subset {{\mathbb {R}}}^2\). Then for any \((y,1) \in S\), \(T(y,1)=(u,1)\) with \(u>y\), and so \(\mathrm{Fix}P_S \circ T = \emptyset \).
5.1 Fixed points of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\): facial characterization
When we take the results of Theorem 5.2 and specify from \(\kappa \) back to the perspective transform \(f^\pi \), we recover the following characterization of the fixed points of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\).
Theorem 5.3
(Facial characterization of fixed points of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\)). Let \(f:X \rightarrow {{\mathbb {R}}}_+ \cup \{+\infty \}\) be a proper closed nonnegative convex function with \(\inf f>0\) and \(\mathrm{argmin}f \ne \emptyset \). Let \(\kappa = f^\pi \). The following hold:
-
(i)
\(\lambda ' = \frac{1}{\min _{u \in X}f(u)},\) where \(\lambda '\) is as in Theorem 5.2(iii);
-
(ii)
Where F is as in Theorem 5.2(i), \(F=\frac{1}{\min _{u \in X}f(u)}(\mathrm{argmin}f \times \{1\})\);
-
(iii)
Any \((x,1) \in \mathrm{Fix}P_S \circ T\) satisfies \(\frac{1+\lambda '}{\lambda '}x \in \mathrm{argmin}f\);
-
(iv)
Any \(y \in \mathrm{argmin}f\) satisfies \((\frac{\lambda '}{1+\lambda '}y,1) \in \mathrm{Fix}P_S \circ T\).
Proof
(i): For simplicity, let \(\eta :=\min _{u \in X}f(u)\). We will first show that
Let \(y \in \mathrm{argmin}f\). Then
and so \((y,1/\eta ) \in D\). To see that \(1/\eta \) is maximal, suppose for a contradiction that there exists \((y_0,\lambda _0) \in \left( X \times \left]1/\eta ,\infty \right[\right) \cap D.\) Then
a contradiction.
(ii): Having shown (i), we have from the definition of F that \(F=D \cap \left\{ (x,1/\eta )\;|\; x \in X \right\} \). Let \((y,1/\eta ) \in F\). Then
and the equality throughout forces \(f(y\eta )=\eta \). Thus \(y\eta \in \mathrm{argmin}f\) and so \(y \in (1/\eta )\mathrm{argmin}f\). Thus \(F \subset (1/\eta )\mathrm{argmin}f\). The reverse inclusion is similar.
(iii) & (iv): By Theorem 5.2(i)a \((x,1) \in \mathrm{Fix}P_S \circ T\) is equivalent to
Having shown (ii), we have that the latter inclusion is equivalent to \(y \eta \in \mathrm{argmin}f\). Combining with (5.7),
Having shown (i), this is equivalent to
which shows both (iii) and (iv). \(\square \)
In the following remark, we compare the facial characterization of fixed points of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) from Theorem 5.3 with the closely related results from [6].
Remark 2
(On synchronicity between Theorems 4.1and5.3). Theorem 5.3 subsumes and is closely connected with the original results of [6, Theorem 7.4], which we recalled as Theorem 4.1. To see why, notice that items (i), (iii), (iv) of Theorem 5.3 have the following characterizations.
(iii) Applying Theorem 4.1(i), we have that \((x,\lambda _*) = T(x,1)\) satisfies \(\lambda _*^{-1}x \in \mathrm{argmin}f\). Theorem 5.2(i) guarantees that \(\lambda _* = \frac{\lambda '}{1+\lambda '}\).
(iv) From Theorem 4.1(ii), \(((1+\eta )^{-1} y,1) \in \mathrm{Fix}P_S \circ T\). From Theorem 5.3(i), \((1+\eta )^{-1} = \frac{\lambda '}{1+\lambda '}\).
(i) The condition \(1/(1+\eta ) = \frac{\lambda '}{1+\lambda '}\) then yields \(\lambda '=1/\eta \).
Theorem 5.3 essentially uses the more general results from Theorem 5.2 to show that the minimizers of f form an exposed face of \((\min _{u \in X}f(u))D\): namely the face that is \((\min _{u \in X}f(u))F\). The other items are all a natural consequence of this.
6 Conclusion
We now state our eponymous convergence result, which shows global convergence of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) in the full generality of [6]. It also, under sufficient conditions to guarantee existence of a fixed point, shows convergence of \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\).
Theorem 6.1
(Convergence of \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) and \({\textbf {GP}}^\mathbf{4}{} {\textbf {A}}\)). Let D be the (closed) fundamental set of \(\kappa \) as in (5.1). Suppose one of the following holds.
-
(i)
\(\kappa = f^\pi \) for \(f:X \rightarrow {{\mathbb {R}}}_+ \cup \left\{ +\infty \right\} \) a proper closed nonnegative convex function with \(\inf f=0\) and \(\mathrm{argmin}f \ne \emptyset \);
-
(ii)
\(\kappa = f^\pi \) for \(f:X \rightarrow {{\mathbb {R}}}_+ \cup \left\{ +\infty \right\} \) a proper closed nonnegative convex function with \(\inf f>0\) and \(\mathrm{argmin}f \ne \emptyset \);
-
(iii)
There exists \(\lambda ' \ge 0\) such that
$$\begin{aligned} \lambda ' = \max _{\lambda \in {{\mathbb {R}}}}\{\lambda \;\;|\;\; \exists y \in X \text {so that}\;(y,\lambda ) \in D \}; \end{aligned}$$ -
(iv)
Such a \(\lambda '\) does not exist and there exists a sequence \((y_n,\lambda _n)_{n \in {{\mathbb {N}}}}\) such that \(\lambda _n\rightarrow \infty \) and \(\lambda _n/\Vert y_n\Vert \rightarrow m>0\).
Let \(\gamma \in [0,1[\) and \((y_0,1) \in S\). Then the following hold.
-
1.
The sequence given by
$$\begin{aligned} (y_{n+1},1):=&\;\mathcal {U}_\gamma (y_{n},1), \nonumber \\ \text {where}\quad \mathcal {U}_\gamma :=&\;(1-\gamma )P_S \circ T +\gamma \mathrm{Id}\end{aligned}$$is convergent to some \((y,1) \in \mathrm{Fix}P_S \circ T\);
-
2.
The shadow sequences \((y_{n+1},\lambda _{n+1})= T(y_n,1)\) satisfy \(\lambda _n \rightarrow \lambda \) for some \(\lambda \in \left[ 0,1\right] \);
-
3.
When (ii) or (iii) holds, \(\lambda = \frac{\lambda '}{1+\lambda '}\);
-
4.
When (ii) holds, \(\lambda ' = 1/(\inf f)\) and \(\left( \frac{1}{\lambda _n}\right) y_n \rightarrow \left( \frac{1+\lambda '}{\lambda '}\right) y \in \mathrm{argmin}f\);
-
5.
When (i) holds, \(y_n \rightarrow y \in \mathrm{argmin}f\) and \(\lambda _n \rightarrow 0\).
Proof
Fixed points: (i): Since \(\inf f=0\), any \(x \in \mathrm{argmin}f\) satisfies
By Theorem 5.3, we have that (ii) \(\implies \) (iii). Either of the assumptions (iii) or (iv) guarantees existence of a fixed point of \(P_S \circ T\) by Theorem 5.2.
Convergence Having shown that a fixed point exists, convergence of \((y_n)_n\) is assured by Corollary 4.12, and the convergence of \((\lambda _n)_n\) is guaranteed by Corollary 4.13. The characterization of \(\lambda '\) in cases (ii) and (iii) is due to Theorems 5.2 and 5.3. \(\square \)
6.1 Further research
We suggest three further avenues of inquiry. Firstly, results on faces of fundamental sets (e.g. Theorem 5.3) are of interest in the development of more general theory. Secondly, Friedlander, Macêdo, and Pong also introduced a second algorithm, \(\mathbf {EMA}\), which is not addressed here [6]. A natural question is whether \(\mathbf {EMA}\) possesses similar properties to \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\). Finally, a motivating question is whether or not algorithms such as \({\textbf {P}}^\mathbf{4}{} {\textbf {A}}\) may have computational advantages for certain problems.
Data Availability Statement
Data availability considerations are not applicable to this research.
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Acknowledgements
The author was supported by Hong Kong Research Grants Council PolyU153085/16p. The author thanks Ting Kei Pong and Michael P. Friedlander for their useful suggestions on this manuscript.
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Lindstrom, S.B. The projected polar proximal point algorithm converges globally. J Glob Optim 84, 177–203 (2022). https://doi.org/10.1007/s10898-022-01136-0
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DOI: https://doi.org/10.1007/s10898-022-01136-0
Keywords
- Projected polar proximal point algorithm
- Gauge optimization
- Polar convolution
- Polar envelope
- Polar proximity operator