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On the local stability of semidefinite relaxations

Abstract

We consider a parametric family of quadratically constrained quadratic programs and their associated semidefinite programming (SDP) relaxations. Given a nominal value of the parameter at which the SDP relaxation is exact, we study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood around the nominal value. Our framework captures a wide array of statistical estimation problems including tensor principal component analysis, rotation synchronization, orthogonal Procrustes, camera triangulation and resectioning, essential matrix estimation, system identification, and approximate GCD. Our results can also be used to analyze the stability of SOS relaxations of general polynomial optimization problems.

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Notes

  1. 1.

    To keep the discussion simple, we are assuming identical and independently distributed Gaussian noise, but many other choices are possible.

  2. 2.

    Although other notions of (set-valued-mapping) continuity exist, they agree for the case of compact valued mappings [32]. Since the analysis done in this paper is local, we may always restrict the range to some closed ball. Hence, we may ignore this distinction in this paper.

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Acknowledgements

We would like to thank Dmitriy Drusvyatskiy for many helpful conversations, and for suggesting the use of the implicit function theorem to analyze our problem. Diego Cifuentes was in the Laboratory for Information and Decision Systems during the development of this paper. Rekha Thomas was partially supported by the NSF grant DMS-1719538. This work was done in part while Pablo Parrilo was visiting the Simons Institute for the Theory of Computing. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF Grant CCF-1740425. This work was also supported in part by the Air Force Office of Scientific Research through AFOSR Grants FA9550-11-1-0305 and the National Science Foundation through NSF Grant CCF-1565235.

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Appendix A. Stability of Lagrange multipliers

Appendix A. Stability of Lagrange multipliers

The goal of this section is to show a generalization of Proposition 4.17 to parametric nonlinear programming. Let \(\Theta \subseteq {\mathbb {R}}^d\) be the parameter space, let \(f : \Theta \times {\mathbb {R}}^N\rightarrow {\mathbb {R}}\) and \(g : \Theta \times {\mathbb {R}}^N\rightarrow {\mathbb {R}}^m\) be continuously differentiable, and such that \(f_\theta ,g_\theta \) are twice continuously differentiable. Consider the parametric family of nonlinear programs

$$\begin{aligned} \min _{x\in \mathbf{X}_\theta } \quad f_\theta (x), \quad \text { where }\quad \mathbf{X}_\theta := \{ x\in {\mathbb {R}}^N: g_\theta (x) = 0 \}. \end{aligned}$$

Let \(L_\theta (x,\lambda ) := f_\theta (x) \!+\! \lambda ^\top g_\theta (x)\) be the Lagrangian function, and let

$$\begin{aligned} \begin{aligned} \mathfrak {L}: \Theta \rightrightarrows {\mathbb {R}}^N\times {\mathbb {R}}^m, \qquad \theta \mapsto \,&\{(x,\lambda ): g_\theta (x) \!=\! 0 ,\; \nabla _{x}L_\theta (x,\lambda ) \!=\! 0\} \end{aligned} \end{aligned}$$

be the Lagrange multiplier mapping. Let \((\bar{x},\bar{\lambda })\in \mathfrak {L}(\bar{\theta })\) be a Lagrange multiplier pair at the nominal parameter \(\bar{\theta }\). We denote \(\bar{H} \!:=\! \frac{1}{2}\nabla ^2_{xx} L_{\bar{\theta }}({\bar{x}},\bar{\lambda })\). We will derive conditions that ensure local stability of \(\mathfrak {L}\) nearby \(\bar{\theta }\). The notion of stability we use is the Aubin property; see [14, 32].

Definition A.1

(Aubin property) Let \(\mathfrak {F}: {\mathbb {R}}^d \rightrightarrows {\mathbb {R}}^n\) be a set-valued mapping. \(\mathfrak {F}\) has the Aubin property at \(\bar{p} \!\in \! {\mathbb {R}}^d\) for \(\bar{y} \!\in \! {\mathbb {R}}^n\) if \(\bar{y} \!\in \! \mathfrak {F}(\bar{p})\) and there is a constant \(\kappa \!\ge \! 0\) and neighborhoods \(U \!\ni \! \bar{y}, V \!\ni \! \bar{p}\) such that

$$\begin{aligned} \mathfrak {F}(p')\cap U \,\subseteq \, \mathfrak {F}(p) + \kappa \, |p'\!-\!p| \,\mathcal {B} \quad \text{ for } \text{ all } p',p\in V, \end{aligned}$$

where \(\mathcal {B} \subseteq {\mathbb {R}}^n\) denotes the unit ball.

The following is the main result of this section.

Theorem A.2

(Stability of Lagrange multipliers) Let \((\bar{x},\bar{\lambda })\in \mathfrak {L}(\bar{\theta })\). Assume that \({\mathrm {ACQ}_{\bar{\mathbf{X}}}(\bar{x})}\) holds, the mapping \(\theta \mapsto \mathbf{X}_\theta \) is smooth nearby \((\bar{\theta },{\bar{x}})\), and \(v^\top {\bar{H}} v \ne 0\) for all nonzero \(v \in T_{\bar{x}}(\mathbf{X}_{\bar{\theta }}) := \ker \nabla g_{\bar{\theta }}({\bar{x}})\). Then the mapping \(\mathfrak {L}\) has the Aubin property at \(\bar{\theta }\) for \((\bar{x},\bar{\lambda })\).

Remark A.3

Similar stability results about Lagrange multipliers appear in the literature (e.g., [8, 17]). However, we were not able to find a result that suited our needs. Previous results either have stronger assumptions (LICQ/MFCQ) or only imply outer semicontinuity of \(\mathfrak {L}\).

The last assumption of Theorem A.2 says that the quadratic form \(v^\top {\bar{H}} v\) is nondegenerate on the tangent space \(T_{{\bar{x}}}(\mathbf{X}_{\bar{\theta }})\). This is similar, but weaker, to the second order sufficient condition for optimality, which states that \(v^\top {\bar{H}} v\) is strictly convex on \(T_{{\bar{x}}}(\mathbf{X}_{\bar{\theta }})\).

Let us see that Theorem A.2 implies Proposition 4.17 from Sect. 4.

Lemma A.4

Let \(\mathfrak {F}:{\mathbb {R}}^d\rightrightarrows {\mathbb {R}}^n\) be a mapping with closed graph. Assume that \(\mathfrak {F}\) has the Aubin property at \(\bar{p}\) for \(\bar{y}\). Then there exists a closed neighborhood \(U_0\ni \bar{y}\) such that \(p\mapsto \mathfrak {F}(p)\cap U_0\) is continuous at \(\bar{p}\).

Proof

From the definition of the Aubin property it is clear that there exists a neighborhood \(U_0\ni \bar{y}\) such that \(\mathfrak {F}\) has the Aubin property at \(\bar{p}\) for y, for any \(y\in U_0 \cap \mathfrak {F}(\bar{p})\). We may assume that \(U_0\) is closed. Let \(\mathfrak {F}_0: p\mapsto \mathfrak {F}(p)\cap U_0\), and note that it has closed graph since \(\mathfrak {F}\) does. Thus, \(\mathfrak {F}_0\) is outer semicontinuous by [32, Thm 5.7]. The lemma follows from [32, Thm 9.38]. \(\square \)

Proof of Proposition 4.17

Since \({\bar{H}} \succeq 0\), then \(v^\top {\bar{H}} v \!=\! 0\) if and only if \({\bar{H}} v \!=\! 0\). Therefore, \(v^\top {\bar{H}} v\) is nondegenerate on \(T_{{\bar{x}}}(\mathbf{X}_{\bar{\theta }})\) if and only if \(\bar{x}\) is not a branch point of \(\bar{\mathbf{X}}\) with respect to \(v \mapsto \bar{H}v\). The proposition follows from Theorem A.2 and Lemma A.4. \(\square \)

We proceed to prove Theorem A.2. The main technical tool we will use is the implicit function theorem, which can be phrased in terms of the Aubin property (see [14, Ex. 4D.3]).

Theorem A.5

(Implicit function) Given \(F : {\mathbb {R}}^{d}\times {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^m\) continuously differentiable, let

$$\begin{aligned} \mathfrak {F}: {\mathbb {R}}^d&\rightrightarrows {\mathbb {R}}^n, \qquad p \mapsto \{y\in {\mathbb {R}}^n: F(p,y)=0\}. \end{aligned}$$

Let \(\bar{p},\bar{y}\) be such that \(\bar{y}\in \mathfrak {F}(\bar{p})\). If \(\nabla _y F (\bar{p},\bar{y})\) is surjective, then \(\mathfrak {F}\) satisfies the Aubin property at \(\bar{p}\) for \(\bar{y}\).

Theorem A.2 would be immediate if \(\mathfrak {L}\) satisfied the hypothesis from Theorem A.5. Unfortunately this is not true, since the defining equations of \(\mathfrak {L}\) may have linearly dependent gradients. In order to fix this problem, we consider a maximal subset of the equations \(g'\subseteq g\) such that \(\{\nabla _x g^i_{\bar{\theta }}(\bar{x})\}_{g^i\in g'}\) are linearly independent. Equivalently, \(g'\subseteq g\) is such that \(\nabla _x g'_{\bar{\theta }}(\bar{x})\) is full rank, and has the same rank as \(\nabla _x g_{\bar{\theta }}(\bar{x})\). Consider the modified solution mapping

$$\begin{aligned} \mathfrak {L}': \theta \mapsto \{(x,\lambda ): {g}_\theta '(x)=0,\, \nabla _{x}L_\theta (x,\lambda ) = 0 \}. \end{aligned}$$

We now apply Theorem A.5 to this new mapping.

Lemma A.6

If \(v^\top {\bar{H}} v \!\ne \! 0\) for all nonzero \(v \!\in \! T_{{\bar{x}}}(\mathbf{X}_{\bar{\theta }})\), then \(\mathfrak {L}'\) has the Aubin property at \(\bar{\theta }\) for \((\bar{x},\bar{\lambda })\).

Proof

Let us see that the surjectivity condition from Theorem A.5 is satisfied. To simplify the notation we will ignore the dependence on \(\theta \), since the only parameter we consider in this proof is \(\bar{\theta }\). Let \(J' \!:=\! \nabla _x g'(\bar{x})\), which is a submatrix of \({J} \!:=\! \nabla _x g(\bar{x})\). By construction, the rows of \(J'\) are linearly independent and \(\ker J' \!=\! \ker J\). Let \(F(x,\lambda ) := (g'(x), \nabla _x L(x,\lambda )\). We need to show that the rows of \(\nabla F(\bar{x},\bar{\lambda })\) are linearly independent. Observe that

$$\begin{aligned} \nabla _{\lambda ,x} F(\bar{x},\bar{\lambda }) = \begin{pmatrix} 0 &{} \quad \nabla _x{g'}(\bar{x})\\ \nabla _x{g}(\bar{x})^\top &{} \quad \nabla ^2_{xx} L({\bar{x}},\bar{\lambda }) \end{pmatrix} = \begin{pmatrix} 0 &{} \quad {J}'\\ {J}^\top &{} \quad 2\bar{H} \end{pmatrix}. \qquad \end{aligned}$$

Let (uv) be a vector in the left kernel of \(\nabla F(\bar{x},\bar{\lambda })\), so that \(v^\top J^\top \!=\! 0\), \(u^\top J' {+} 2 v^\top \bar{H} \!=\! 0\). We need to show that \((u,v)\!=\!0\). Since \(v \!\in \! \ker J \!=\! \ker J'\) then \(0 = (u^\top J' {+} 2 v^\top \bar{H}) v = 2 v^\top \bar{H} v\). As \(v \in \ker J = T_{\bar{x}}(\bar{\mathbf{X}})\) and \({\bar{v}}^\top \bar{H} v \!=\! 0\), then \(v \!=\! 0\) by the assumption. Therefore \(0 = u^\top J' {+} 2 v^\top \bar{H} = u^\top J'\), and thus \(u {=} 0\) since the rows of \(J'\) are linearly independent. \(\square \)

In order to prove Theorem A.2 it remains to see that the modified mapping \(\mathfrak {L}'\) agrees with \(\mathfrak {L}\), at least locally. This follows from the following lemma.

Lemma A.7

Let \(\mathbf{X}_\theta \subseteq {\mathbf{X}_\theta '}\subseteq {\mathbb {R}}^N\) be the zero sets of \(g_\theta , {g_\theta '}\). Assume that \({\mathrm {ACQ}_{\bar{\mathbf{X}}}(\bar{x})}\) holds, and that the mapping \(\theta \mapsto \mathbf{X}_\theta \) is smooth nearby \((\bar{\theta },{\bar{x}})\). Then there are neighborhoods \(V_0\ni \bar{\theta }\) and \(U_0\ni \bar{x}\) such that \( \mathbf{X}_\theta \cap U_0 = \mathbf{X}_\theta ' \cap U_0\) for all \(\theta \in V_0\).

The proof of Lemma A.7 requires an auxiliary lemma.

Lemma A.8

Let \( \mathbf{W}:= \{w\!\in \!{\mathbb {R}}^K: g(w){=}0\},\) where \(g=(g^1,\dots ,g^m)\), and assume that \(\mathbf{W}\) is a smooth D-dimensional manifold nearby \(\bar{w}\). Let \(g'=(g^1,\ldots ,g^{K-D})\subseteq g\) be such that their gradients at \(\bar{w}\) are linearly independent. Then there is a neighborhood \(U\subseteq {\mathbb {R}}^K\) of \(\bar{w}\) such that \(\mathbf{W}\cap U = \mathbf{W}'\cap U\), where \(\mathbf{W}':=\{w: g'(w)=0\}\).

Proof

By the implicit function theorem \(\mathbf{W}'\) is a D-dimensional manifold nearby \(\bar{w}\). Thus, there is a neighborhood \(U\subseteq {\mathbb {R}}^K\) of \(\bar{w}\) such that \(\mathbf{W}\cap U\) is a submanifold of \(\mathbf{W}'\cap U\). Since they have the same dimension, \(\mathbf{W}\cap U\) must be an open set of \(\mathbf{W}'\cap U\). \(\square \)

Proof of Lemma A.8

Let \(\mathbf{W}:=\{(\theta ,x): g_\theta (x){=}0\}\) and \(\mathbf{W}':=\{(\theta ,x): g_\theta '(x){=}0\}\). We will use Lemma A.8 to show the existence of a neighborhood \(U \!\ni \! {\bar{w}}\), such that \(\mathbf{W}\cap U \!=\! \mathbf{W}'\cap U\). Note that this would conclude the proof. By assumption we know that \(\mathbf{W}\) is a smooth manifold nearby \({\bar{w}}:= (\bar{x},\bar{\theta })\) of dimension \(D:= \dim \Theta \!+\! \dim _{\bar{x}} \bar{\mathbf{X}} \). Recall that by construction of \(g'\) the gradients \(\{\nabla g^i_{\bar{\theta }}(\bar{x})\}_{g^i \in g'}\) are linearly independent, and the number of equations is \(|g'| = {{\,\mathrm{rank}\,}}\nabla g_{\bar{\theta }}({\bar{x}})\). Since \({\mathrm {ACQ}_{\bar{\mathbf{X}}}({\bar{x}})}\) holds, then

$$\begin{aligned} |g'| = {{\,\mathrm{rank}\,}}\nabla g_{\bar{\theta }}({\bar{x}}) = N - \dim _{{\bar{x}}} \bar{\mathbf{X}} = (\dim \Theta + N) - D. \end{aligned}$$

So the assumptions of Lemma A.8 are satisfied, as wanted. \(\square \)

Proof of Theorem A.2

The Aubin property is a local condition. Since \(\mathfrak {L},\mathfrak {L}'\) agree nearby \(\bar{\theta },\bar{x}\) (Lemma A.7), and since \(\mathfrak {L}'\) has the Aubin property (Lemma A.6), then the same holds for \(\mathfrak {L}\). \(\square \)

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Cifuentes, D., Agarwal, S., Parrilo, P.A. et al. On the local stability of semidefinite relaxations. Math. Program. (2021). https://doi.org/10.1007/s10107-021-01696-1

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Keywords

  • Parametric SDP
  • Stability
  • Sum of squares
  • Algebraic variety

Mathematics Subject Classification

  • Primary: 90C22
  • Secondary: 90C31