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Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping

Abstract

The existing algorithms for solving the convex minimization problem over the fixed point set of a nonexpansive mapping on a Hilbert space are based on algorithmic methods, such as the steepest descent method and conjugate gradient methods, for finding a minimizer of the objective function over the whole space, and attach importance to minimizing the objective function as quickly as possible. Meanwhile, it is of practical importance to devise algorithms which converge in the fixed point set quickly because the fixed point set is the set with the constraint conditions that must be satisfied in the problem. This paper proposes an algorithm which not only minimizes the objective function quickly but also converges in the fixed point set much faster than the existing algorithms and proves that the algorithm with diminishing step-size sequences strongly converges to the solution to the convex minimization problem. We also analyze the proposed algorithm with each of the Fletcher–Reeves, Polak–Ribiére–Polyak, Hestenes–Stiefel, and Dai–Yuan formulas used in the conventional conjugate gradient methods, and show that there is an inconvenient possibility that their algorithms may not converge to the solution to the convex minimization problem. We numerically compare the proposed algorithm with the existing algorithms and show its effectiveness and fast convergence.

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Notes

  1. \((d_n^f)_{n\in \mathbb {N}}\) is referred to as a descent search direction if \(\langle d_n^f, {\nabla }\! f (x_n) \rangle < 0\) for all \(n\in \mathbb {N}\).

  2. These are defined as follows: \(\delta _n^{\mathrm {FR}}:=\Vert {\nabla }\! f (x_{n+1})\Vert ^2 /\Vert {\nabla }\! f (x_n)\Vert ^2, \delta _n^{\mathrm {PRP}}:= v_n/\Vert {\nabla }\! f (x_n)\Vert ^2, \delta _n^{\mathrm {HS}}:= v_n / u_n, \delta _n^{\mathrm {DY}}:=\Vert \nabla f (x_{n+1})\Vert ^2 /u_n\), where \(u_n := \langle d_n^f, {\nabla }\! f (x_{n+1}) - {\nabla }\! f (x_n) \rangle \) and \(v_n:= \langle {\nabla }\! f (x_{n+1}), {\nabla }\! f (x_{n+1}) - {\nabla }\! f (x_n) \rangle \).

  3. For example, when there is a bound on \(\mathrm {Fix}(N)\), we can choose \(K\) as a closed ball with a large radius containing \(\mathrm {Fix}(N)\). The metric projection onto such a \(K\) is easily computed (see also Sect. 2.1). See the final paragraph in Sect. 3.1 for a discussion of Problem 3.1 when a bound on \(\mathrm {Fix}(N)\) either does not exist or is not known.

  4. The conjugate gradient method with the DY formula (i.e., \(\delta _n^{(1)}:= \delta _n^{\mathrm {DY}}\)) generates the descent search direction under the Wolfe conditions [29]. Whether or not the conjugate gradient methods generate descent search directions depends on the choices of \(\delta _n^{(1)}\) and \(\alpha _n\).

  5. Reference [10], Sect. 2.1] showed that \(x_{n+1}:= x_n + \alpha _n d_n^{f}\) and \(d_{n+1}^{f}:= - {\nabla }\! f (x_{n+1}) + \delta _n^{(1)} d_n^{f} - \delta _n^{(2)} z_n\), where \(\alpha _n, \delta _n^{(1)} (>0)\) are arbitrary, \(z_n (\in \mathbb {R}^N)\) is any vector, and \(\delta _n^{(2)}:= \delta _n^{(1)} (\langle \nabla f(x_{n+1}), d_n \rangle / \langle {\nabla }\! f(x_{n+1}), z_n \rangle )\), satisfy \(\langle d_n^f, {\nabla }\! f(x_n) \rangle = -\Vert {\nabla }\! f(x_n)\Vert ^2 (n\in \mathbb {N})\).

  6. We can choose, for example, \(w_n:= N(y_n) - y_n\) and \(z_n:= {\nabla }\! f(x_{n+1}) (n\in \mathbb {N})\) by referring to [12] and [15], Sect. 3]. Lemma 3.1 ensures that they are bounded.

  7. Given a halfspace \(S:= \{ x\in H :\langle a,x\rangle \le b \}\), where \(a (\ne 0) \in H\) and \(b\in \mathbb {R}, N (x):= P_{S} (x) = x - [\max \{ 0, \langle a,x \rangle -b \} /\Vert a\Vert ^2] a (x\in H)\) is nonexpansive with \(\mathrm {Fix}(N) = \mathrm {Fix}(P_{S}) = S \ne \emptyset \) [18, p. 406], [17], Chap. 28.3]. However, we cannot define a bounded \(K\) satisfying \(\mathrm {Fix}(N) = S \subset K\).

  8. Suppose that \((x_n)_{n\in \mathbb {N}} (\subset H)\) weakly converges to \(\hat{x} \in H\) and \(\bar{x} \ne \hat{x}\). Then, the following condition, called Opial’s condition [30], is satisfied: \(\liminf _{n\rightarrow \infty }\Vert x_n - \hat{x}\Vert < \liminf _{n\rightarrow \infty }\Vert x_n - \bar{x}\Vert \). In the above situation, Opial’s condition leads to \(\liminf _{i \rightarrow \infty }\Vert x_{n_i} - x^*\Vert < \liminf _{i \rightarrow \infty }\Vert x_{n_i} - \hat{N} (x^*)\Vert \).

  9. We randomly chose \(\lambda _Q^k \!\in \! (1, S) (k\!=\!2,3,\ldots , S\!-\!1)\) and set \(\hat{Q} \!\in \! \mathbb {R}^{S \times S}\) as a diagonal matrix with eigenvalues, \(\lambda _Q^1, \lambda _Q^2, \ldots , \lambda _Q^S\). We made a positive definite matrix, \(Q \!\in \! \mathbb {R}^{S \times S}\), using an orthogonal matrix and \(\hat{Q}\).

  10. \(x\in \mathbb {R}^S\) satisfies \(\Vert x - N(x)\Vert = 0\) if and only if \(x\in \mathrm {Fix}(N)\).

  11. See Remark 3.2 on the nonmonotonicity of \((\Vert x_n - N(x_n)\Vert )_{n\in \mathbb {N}}\) in Algorithm 3.1.

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Acknowledgments

I wrote Sect. 3.2 by referring to the referee’s report on the original manuscript of [14]. I am sincerely grateful to the anonymous referee that reviewed the original manuscript of [14] for helping me compile the paper. I also would like to thank the Co-Editor, Michael C. Ferris, and the two anonymous reviewers for helping me improve the original manuscript.

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Correspondence to Hideaki Iiduka.

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This work was supported by the Japan Society for the Promotion of Science through a Grant-in-Aid for Young Scientists (B) (23760077), and in part by the Japan Society for the Promotion of Science through a Grant-in-Aid for Scientific Research (C) (22540175).

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Iiduka, H. Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping. Math. Program. 149, 131–165 (2015). https://doi.org/10.1007/s10107-013-0741-1

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Keywords

  • Convex optimization
  • Fixed point set
  • Nonexpansive mapping
  • Conjugate gradient method
  • Three-term conjugate gradient method
  • Fixed point optimization algorithm

Mathematics Subject Classification (2010)

  • 47H07
  • 47H09
  • 65K05
  • 65K10
  • 90C25
  • 90C30
  • 90C52