Characterizations of the Aubin property of the solution mapping for nonlinear semidefinite programming

For a locally optimal solution to the nonlinear semidefinite programming problem, under Robinson’s constraint qualification, the following conditions are proved to be equivalent: the strong second-order sufficient condition and constraint nondegeneracy; the nonsingularity of Clarke’s Jacobian of the Karush-Kuhn-Tucker system; the strong regularity of the Karush-Kuhn-Tucker point; and others.

Recently, Chan and Sun [Chan, Z. X., D. Sun. Constraint nondegeneracy, strong regularity and nonsingularity in semidefinite programming. SIAM J. Optim. 19 370–376.] reported for semidefinite programming (SDP) that the primal/dual constraint nondegeneracy is equivalent to the dual/primal strong second order sufficient condition (SSOSC). This result is responsible for a number of important results in stability analysis of SDP. In this paper, we study duality of this type in nonlinear semidefinite programming (NSDP). We introduce the dual SSOSC at a Karush-Kuhn-Tucker (KKT) triple of NSDP and study its various characterizations and relationships to the primal nondegeneracy. Although the dual SSOSC is nothing but the SSOSC for the Wolfe dual of the NSDP, it suggests new information for the primal NSDP. For example, it ensures that the inverse of the Hessian of the Lagrangian function exists at the KKT triple and the inverse is positive definite on some normal space. It also ensures the primal nondegeneracy. Some of our results generalize the corresponding classical duality results in nonlinear programming studied by Fujiwara et al. [Fujiwara, O., S.-P. Han, O. L. Mangasarian. 1984. Local duality of nonlinear programs. SIAM J. Control Optim. 22 162–169]. For the convex quadratic SDP (QSDP), we have complete characterizations for the primal and dual SSOSC. Our results reveal that the nearest correlation matrix problem satisfies not only the primal and dual SSOSC but also the primal and dual nondegeneracy at its solution, suggesting that it is a well-conditioned QSDP.

Geometric methods are important for researching the differential properties of metric projectors, sensitivity analysis, and the augmented Lagrangian algorithm. Sun [3] researched the relationship among the strong second-order sufficient condition, constraint nondegeneracy, B-subdifferential nonsingularity of the KKT system, and the strong regularity of KKT points in investigating nonlinear semidefinite programming problems. Geometric properties of cones are necessary in studying second-order sufficient condition and constraint nondegeneracy. In this paper, we study the geometric properties of a class of nonsymmetric cones, which is widely applied in optimization problems subjected to the epigraph of vector k-norm functions and low-rank-matrix approximations. We compute the polar, the tangent cone, the linear space of the tangent cone, the critical cone, and the affine hull of this critical cone. This paper will support future research into the sensitivity and algorithms of related optimization problems.

In this paper, we consider a cone problem of matrix optimization induced by spectral norm (MOSN). By Schur complement, MOSN can be reformulated as a nonlinear semidefinite programming (NLSDP) problem. Then we discuss the constraint nondegeneracy conditions and strong second-order sufficient conditions of MOSN and its SDP reformulation, and obtain that the constraint nondegeneracy condition of MOSN is not always equivalent to that of NLSDP. However, the strong second-order sufficient conditions of these two problems are equivalent without any assumption. Finally, a sufficient condition is given to ensure the nonsingularity of the Clarke’s generalized Jacobian of the KKT system for MOSN.

Exact SDP relaxations for classes of nonlinear semidefinite programming problems

In this paper, we introduce a transformation that converts a class of linear and nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems. For those problems of interest, the transformation replaces matrix-valued constraints by vector-valued ones, hence reducing the number of constraints by an order of magnitude. The class of transformable problems includes instances of SDP relaxations of combinatorial optimization problems with binary variables as well as other important SDP problems. We also derive gradient formulas for the objective function of the resulting nonlinear optimization problem and show that both function and gradient evaluations have affordable complexities that effectively exploit the sparsity of the problem data. This transformation, together with the efficient gradient formulas, enables the solution of very large-scale SDP problems by gradient-based nonlinear optimization techniques. In particular, we propose a first-order log-barrier method designed for solving a class of large-scale linear SDP problems. This algorithm operates entirely within the space of the transformed problem while still maintaining close ties with both the primal and the dual of the original SDP problem. Global convergence of the algorithm is established under mild and reasonable assumptions.

In this work, we derive second-order optimality conditions for nonlinear semidefinite programming (NSDP) problems, by reformulating it as an ordinary nonlinear programming problem using squared slack variables. We first consider the correspondence between Karush-Kuhn-Tucker points and regularity conditions for the general NSDP and its reformulation via slack variables. Then, we obtain a pair of "no-gap" second-order optimality conditions that are essentially equivalent to the ones already considered in the literature. We conclude with the analysis of some computational prospects of the squared slack variables approach for NSDP.

We derive a new genericity result for nonlinear semidefinite programming (NLSDP). Namely, almost all linear perturbations of a given NLSDP are shown to be nondegenerate. Here, nondegeneracy for NLSDP refers to the transversality constraint qualification, strict complementarity and second-order sufficient condition. Due to the presence of the second-order sufficient condition, our result is a nontrivial extension of the corresponding results for linear semidefinite programs (SDP) from Alizadeh et al. (Math Program 77(2, Ser. B):111---128, 1997). The proof of the genericity result makes use of Forsgren's derivation of optimality conditions for NLSDP in Forsgren (Math Program 88(1, Ser. A):105---128, 2000). Due to the latter approach, the positive semidefiniteness of a symmetric matrix G(x), depending continuously on x, is locally equivalent to the fact that a certain Schur complement S(x) of G(x) is positive semidefinite. This yields a reduced NLSDP by considering the new semidefinite constraint $$S(x) \succeq 0$$S(x)ź0, instead of $$G(x) \succeq 0$$G(x)ź0. While deriving optimality conditions for the reduced NLSDP, the well-known and often mentioned "H-term" in the second-order sufficient condition vanishes. This allows us to access the proof of the genericity result for NLSDP.

In this paper, we study augmented Lagrangian functions for nonlinear semidefinite programming (NSDP) problems with exactness properties. The term exact is used in the sense that the penalty parameter can be taken appropriately, so a single minimization of the augmented Lagrangian recovers a solution of the original problem. This leads to reformulations of NSDP problems into unconstrained nonlinear programming ones. Here, we first establish a unified framework for constructing these exact functions, generalizing Di Pillo and Lucidi's work from 1996, that was aimed at solving nonlinear programming problems. Then, through our framework, we propose a practical augmented Lagrangian function for NSDP, proving that it is continuously differentiable and exact under the so-called nondegeneracy condition. We also present some preliminary numerical experiments.

It was proved in Izmailov and Solodov (2014). Newton-Type Methods for Optimization and Variational Problems, Springer] that the existence of a noncritical multiplier for a (smooth) nonlinear programming problem is equivalent to an error bound condition for the Karush–Kuhn–Thcker (KKT) system without any assumptions. This paper investigates whether this result still holds true for a (smooth) nonlinear semidefinite programming (SDP) problem. The answer is negative: the existence of noncritical multiplier does not imply the error bound condition for the KKT system without additional conditions, which is illustrated by an example. In this paper, we obtain characterizations, in terms of the problem data, the critical and noncritical multipliers for a SDP problem. We prove that, for the SDP problem, the noncriticality property can be derived from the error bound condition for the KKT system without any assumptions, and we give an example to show that the noncriticality does not imply the error bound for the KKT system. We propose a set of assumptions under which the error bound condition for the KKT system can be derived from the noncriticality property. a Finally, we establish a new error bound for [Formula: see text]-part, which is expressed by both perturbation and the multiplier estimation.

In this paper, an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming (SDP) problems is described and analyzed. Such nonlinear and nonconvex programs arise, e.g., in the design of optimal static or reduced order output feedback control laws and have the structure of abstract optimal control problems in a finite dimensional Hilbert space. The algorithm treats the abstract states and controls as independent variables. In particular, an algorithm for minimizing a nonlinear matrix objective functional subject to a nonlinear SDP-condition, a positive definiteness condition, and a nonlinear matrix equation is considered. The algorithm is designed to take advantage of the structure of the problem. It is an extension of an interior point trust region method to nonlinear and nonconvex SDPs, with a special structure which applies sequential quadratic programming techniques to a sequence of barrier problems and uses trust regions to ensure robustness of the iteration. Some convergence results are given, and, finally, several numerical examples demonstrate the applicability of the considered algorithm.

ABSTRACTThe proximal method of multipliers was proposed by Rockafellar [Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math Oper Res. 1976;1:97–116] for solving convex programming and it is a kind of proximal point method applied to convex programming. In this paper, we apply this method for solving nonlinear semidefinite programming problems, in which subproblems have better properties than those from the augmented Lagrange method. We prove that, under the linear independence constraint qualification and the strong second-order sufficiency optimality condition, the rate of convergence of the proximal method of multipliers, for a nonlinear semidefinite programming problem, is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold . Moreover, the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to .

We characterize the local upper Lipschitz property of the stationary point mapping and the Karush–Kuhn–Tucker (KKT) mapping for a nonlinear second-order cone programming problem using the graphical derivative criterion. We demonstrate that the second-order sufficient condition and the strict constraint qualification are sufficient for the local upper Lipschitz property of the stationary point mapping and are both sufficient and necessary for the local upper Lipschitz property of the KKT mapping.

This paper is devoted to a new approach for solving nonlinear programming (NLP) problems for which the Kuhn-Tucker optimality conditions system of equations is singular. It happens when the strict complementarity condition (SCC), a constrained qualification (CQ), and a second-order sufficient condition (SOSC) for optimality is not necessarily satisfied at a solution. Our approach is based on the construction of p-regularity and on reformulating the inequality constraints as equality. Namely, by introducing the slack variables, we get the equality constrained problem, for which the Lagrange optimality system is singular at the solution of the NLP problem in the case of the violation of the CQs, SCC and/or SOSC. To overcome the difficulty of singularity, we propose the p-factor method for solving the Lagrange system. The method has a superlinear rate of convergence under a mild assumption. We show that our assumption is always satisfied under a standard second-order sufficient optimality condition.

A sequentially semidefinite programming method is proposed for solving nonlinear semidefinite programming problem (NLSDP). Inspired by the sequentially quadratic programming (SQP) method, the algorithm generates a search direction by solving a quadratic semidefinite programming subproblem at each iteration. The [Formula: see text] exact penalty function and a line search strategy are used to determine whether the trial step can be accepted or not. Under mild assumptions, the proposed algorithm is globally convergent. In order to avoid the Maratos effect, we present a modified SQP-type algorithm with the second-order correction step and prove that the fast local superlinear convergence can be obtained under the strict complementarity and the second-order sufficient condition with the sigma term. Finally, some numerical experiments are given to show the effectiveness of the algorithm.