Cone Complementarity Constraints Research Articles

New Constraint Qualifications for Mathematical Programs with Second-Order Cone Complementarity Constraints

New Constraint Qualifications for Mathematical Programs with Second-Order Cone Complementarity Constraints

Scaled constraint qualifications and necessary optimality conditions for nonsmooth mathematical programs with second-order cone complementarity constraints

Scaled constraint qualifications and necessary optimality conditions for nonsmooth mathematical programs with second-order cone complementarity constraints

Second-order Optimality Conditions for Mathematical Program with Semidefinite Cone Complementarity Constraints and Applications

Second-order Optimality Conditions for Mathematical Program with Semidefinite Cone Complementarity Constraints and Applications

Semidefinite relaxation method for polynomial optimization with second-order cone complementarity constraints

<p style='text-indent:20px;'>Polynomial optimization problem with second-order cone complementarity constraints (SOCPOPCC) is a special case of mathematical program with second-order cone complementarity constraints (SOCMPCC). In this paper, we consider how to apply Lasserre's type of semidefinite relaxation method to solve SOCPOPCC. To this end, we first reformulate SOCPOPCC equivalently as a polynomial optimization and then solve the reformulated polynomial optimization with semidefinite relaxation method. For a special case of SOCPOPCC, we present another reformulation of polynomial optimization, which is of lower degree. SDP relaxation method is applied to solve the new polynomial optimization. Numerical examples are reported to show the efficiency of our proposed method.</p>

On the calm b-differentiability of projector onto circular cone and its applications

In this paper, we study a concept on the calm B-differentiability, a new kind of generalized differentiabilities for a given vector function introduced by Ye and Zhou in 2017, of the projector onto the circular cone. Then, we discuss its applications in mathematical programming problems with circular cone complementarity constraints. Here, this problem can be considered to be a generalization of mathematical programming problems with second-order cone complementarity constraints, and thus it includes a large class of mathematical models in optimization theory. Consequently, the obtained results for this problem are generalized, and then corresponding results for some special mathematical problems can be implied from them directly. For more detailed information, we will first prove the calmly B-differentiable property of the projector onto the circular cone. This result is not easy to be shown by simply resorting to those of the projection operator onto the second-order cone. By virtue of exploiting variational techniques, we next establish the exact formula for the regular (Fréchet) normal cone (this concept was proposed by Kruger and Mordukhovich in 1980) to the circular cone complementarity set. Note that this set can be considered to be a generalization of the second-order cone complementarity set. In finally, the exact formula for the regular (Fréchet) normal cone to the circular cone complementarity set would be useful for us to study first-order necessary optimality conditions for mathematical programming problems with circular cone complementarity constraints. Our obtained results in the paper are new, and they are generalized to some existing ones in the literature.

Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints

A stochastic mathematical program model with second-order cone complementarity constraints (SSOCMPCC) is introduced in this paper. It can be considered as a non-trivial extension of stochastic mathematical program with complementarity constraints, and could arise from a hard-to-handle class of bilivel second-order cone programming and inverse stochastic second-order cone programming. By introducing the Chen-Harker-Kanzow-Smale (CHKS) type function to replace the projection operator onto the second-order cone, a smoothing sample average approximation (SAA) method is proposed for solving the SSOCMPCC problem. It can be shown that with proper assumptions, as the sample size goes to infinity, any cluster point of global solutions of the smoothing SAA problem is a global solution of SSOCMPCC almost surely, and any cluster point of stationary points of the former problem is a C-stationary point of the latter problem almost surely. C-stationarity can be strengthened to M-stationarity with additional assumptions. Finally, we report a simple illustrative numerical test to demonstrate our theoretical results.

Mathematical Programs with Second-Order Cone Complementarity Constraints: Strong Stationarity and Approximation Method

The existence of complementarity constraints causes the difficulties for studying mathematical programs with second-order cone complementarity constraints, since the standard constraint qualification, such as Robinson’s constraint qualification, is invalid. Therefore, various stationary conditions including strong, Mordukhovich and Clarke stationary conditions have been proposed, according to different reformulations of the second-order cone complementarity constraints. In this paper, we present a new reformulation of this problem by taking into consideration the Jordan algebra associated with the second-order cone. It ensures that the classical Karush–Kuhn–Tucker condition coincides with the strong stationary condition of the original problem. Furthermore, we propose a class of approximation methods to solve mathematical programs with second-order cone complementarity constraints. Any accumulation point of the iterative sequences, generated by the approximation method, is Clarke stationary under the corresponding linear independence constraint qualification. This stationarity can be enhanced to strong stationarity with an extra strict complementarity condition. Preliminary numerical experiments indicate that the proposed method is effective.

Exact Formula for the Second-Order Tangent Set of the Second-Order Cone Complementarity Set

The second-order tangent set is an important concept in describing the curvature of the set involved. Due to the existence of the complementarity condition, the second-order cone (SOC) complementarity set is a nonconvex set. Moreover, unlike the vector complementarity set, the SOC complementarity set is not even the union of finitely many polyhedral convex sets. Despite these difficulties, we succeed in showing that like the vector complementarity set, the SOC complementarity set is second-order directionally differentiable and an exact formula for the second-order tangent set of the SOC complementarity set can be given. We derive these results by establishing the relationship between the second-order tangent set of the SOC complementarity set and the second-order directional derivative of the projection operator over the second-order cone, and calculating the second-order directional derivative of the projection operator over the second-order cone. As an application, we derive second-order necessary optimality conditions for the mathematical program with second-order cone complementarity constraints.

On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints

This paper focuses on a class of mathematical programs with symmetric cone complementarity constraints (SCMPCC). The explicit expression of C-stationary condition and SCMPCC-linear independence constraint qualification (denoted by SCMPCC-LICQ) for SCMPCC are first presented. We analyze a parametric smoothing approach for solving this program in which SCMPCC is replaced by a smoothing problem \begin{document}$P_{\varepsilon}$\end{document} depending on a (small) parameter \begin{document}$\varepsilon$\end{document} . We are interested in the convergence behavior of the feasible set, stationary points, solution mapping and optimal value function of problem \begin{document}$P_{\varepsilon}$\end{document} when \begin{document}$\varepsilon \to 0$\end{document} under SCMPCC-LICQ. In particular, it is shown that the convergence rate of Hausdorff distance between feasible sets \begin{document}$\mathcal{F}_{\varepsilon}$\end{document} and \begin{document}$\mathcal{F}$\end{document} is of order \begin{document}$\mbox{O}(|\varepsilon|)$\end{document} and the solution mapping and optimal value of \begin{document}$P_{\varepsilon}$\end{document} are outer semicontinuous and locally Lipschitz continuous at \begin{document}$\varepsilon=0$\end{document} respectively. Moreover, any accumulation point of stationary points of \begin{document}$P_{\varepsilon}$\end{document} is a C-stationary point of SCMPCC under SCMPCC-LICQ.

A Sequential Convex Program Approach to an Inverse Linear Semidefinite Programming Problem

This paper is devoted to the study of solving method for a type of inverse linear semidefinite programming problem in which both the objective parameter and the right-hand side parameter of the linear semidefinite programs are required to adjust. Since such kind of inverse problem is equivalent to a mathematical program with semidefinite cone complementarity constraints which is a rather difficult problem, we reformulate it as a nonconvex semi-definte programming problem by introducing a nonsmooth partial penalty function to penalize the complementarity constraint. The penalized problem is actually a nonsmooth DC programming problem which can be solved by a sequential convex program approach. Convergence analysis of the penalty models and the sequential convex program approach are shown. Numerical results are reported to demonstrate the efficiency of our approach.

Exact formulas for the proximal/regular/limiting normal cone of the second-order cone complementarity set

The proximal, regular and limiting normal cones to the second-order cone complementarity set play important roles in studying mathematical programs with second-order cone complementarity constraints, second-order cone programs, and the second-order cone complementarity problems. It is needed in the first-order optimality conditions for mathematical programs with second-order cone complementarity constraint, the second-order subdifferential criteria in characterizing the full stability for second-order cone programs and second-order cone complementarity problems, as well as in the characterizing the pseudo-Lipschitz continuity of the solution mapping to parametric second-order cone complementarity problems. In this paper we establish explicit formulas for the proximal, regular, and limiting normal cone of the second-order cone complementarity set.

First-Order Optimality Conditions for Mathematical Programs with Second-Order Cone Complementarity Constraints

In this paper we consider a mathematical program with second-order cone complementarity constraints (SOCMPCC). The SOCMPCC generalizes the mathematical program with complementarity constraints (MPCC) in replacing the set of nonnegative reals by second-order cones. There are difficulties in applying the classical Karush--Kuhn--Tucker (KKT) condition to the SOCMPCC directly since the usual constraint qualification such as Robinson's constraint qualification never holds if it is considered as an optimization problem with a convex cone constraint. Using various reformulations and recent results on the exact formula for the proximal/regular and limiting normal cone, we derive necessary optimality conditions in the forms of the strong, Mordukhovich, and Clarke (resp., S, M, and C) stationary conditions under certain constraint qualifications. We also show that, unlike the MPCC, the classical KKT condition of the SOCMPCC is in general not equivalent to the S-stationary condition unless the dimension of each seco...

First order necessary optimality conditions for mathematical programs with second-order cone complementarity constraints

This paper is to develop first order necessary optimality conditions for a mathematical program with second-order cone complementarity constraints (MPSCC) which includes the mathematical program with (vector) complementarity constraints (MPCC) as a special case. Like the case of MPCC, Robinson's constraint qualification fails at every feasible point of MPSCC if we treat the MPSCC as an ordinary optimization problem. Using the formulas of regular and limiting coderivatives and generalized Clarke's Jacobian of the projection operator onto second-order cones from the literature, we present the S-, M-, C- and A-stationary conditions for a MPSCC problem. Moreover, several constraint qualifications including MPSCC-Abadie CQ, MPSCC-LICQ, MPSCC-MFCQ and MPSCC-GMFCQ are proposed, under which a local minimizer of MPSCC is shown to be a S-, M-, C- or A-stationary point.

Variational Geometry of the Complementarity Set for Second Order Cone

The Mordukhovich coderivative has become an important tool for stability and optimality in variational analysis. In this paper, we focus on variational geometry of the complementarity set for second order cone. Particularly, the tangent cone and the normal cone to the complementarity set for second order cone are explicitly characterized. The result developed in this paper paves the way for characterizing the Mordukhovich coderivative of the normal mapping in the study of the Aubin property and optimality conditions for mathematical programs with second order cone complementarity constraints.

On Properties of the Bilinear Penalty Function Method for Mathematical Programs with Semidefinite Cone Complementarity Constraints

This paper is devoted to the study of properties of the bilinear penalty function method for the mathematical programs with semidefinite cone complementarity constraints(SDCMPCC), in which the complementarity constraint appears as a bilinear penalty term in the objective. We discuss the relationships of solutions between the SDCMPCC and its bilinear penalty problem as well as the second order sufficient conditions and the constraint qualifications between these two problems. Moreover, under certain conditions, we demonstrate that any accumulation point of the sequence generated by the bilinear penalty function method is an S-stationary point of the SDCMPCC.

Two approaches for solving mathematical programs with second-order cone complementarity constraints

This paper considers a mathematical program with second-order cone complementarity constrains (MPSOCC). We present two approximation methods for solving the MPSOCC. One employs some smoothing functions to approximate the MPSOCC and the other makes use of some techniques to relax the complementarity constrains in the MPSOCC. We investigate the limiting behavior of both methods. In particular, we show that, under mild conditions, any accumulation point of stationary points of the approximation problems must be a Clarke-type stationary point of the MPSOCC.

A perturbation approach for an inverse quadratic programming problem over second-order cones

This paper is devoted to studying a type of inverse second-order cone quadratic programming problems, in which the parameters in both the objective function and the constraint set of a given second-order cone quadratic programming problem need to be adjusted as little as possible so that a known feasible solution becomes optimal. This inverse problem can be written as a minimization problem with second-order cone complementarity constraints and a positive semidefinite cone constraint. Applying the duality theory, we reformulate this problem as a linear second-order cone complementarity constrained optimization problem with a semismoothly differentiable objective function, which has fewer variables than the original one. A perturbed problem is proposed with the help of the projection operator over second-order cones, whose feasible set and optimal solution set are demonstrated to be continuous and outer semicontinuous, respectively, as the parameter decreases to zero. A smoothing Newton method is constructed to solve the perturbed problem and its global convergence and local quadratic convergence rate are shown. Finally, the numerical results are reported to show the effectiveness for the smoothing Newton method to solve the inverse second-order cone quadratic programming problem.

First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We first derive explicit formulas for the proximal and limiting normal cone of the graph of the normal cone to the positive semidefinite cone. Using these formulas and classical nonsmooth first order necessary optimality conditions we derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions. Moreover we give constraint qualifications under which a local solution of SDCMPCC is a S-, M- and C-stationary point. Moreover we show that applying these results to MPCC produces new and weaker necessary optimality conditions.

Necessary Optimality Conditions for Mathematical Programs with Second-Order Cone Complementarity Constraints

This paper considers the mathematical program with second-order cone complementarity constrains (MPSOCC). As a generalization of the developed mathematical program with complementarity constrains (MPCC), MPSOCC has many applications in practice. Motivated by the MPCC theory, several stationarity concepts, which include the Clarke-type, Mordukhovich-type, and strong stationarities, are presented in this paper. It is further shown that a local minimizer of MPSOCC must be stationary in some sense under suitable conditions. This indicates that these stationarity concepts are reasonable in theory.

Mathematical Programs with Semidefinite Cone Complementarity Constraints: Constraint Qualifications and Optimality Conditions

The tangent cone of gph\(N_{S^n_+}\) plays an important role in developing necessary conditions for mathematical programs with semidefinite cone complementarity constraints. We demonstrate an elegant formula for the tangent cone of gph\(N_{S^n_+}\), based on which the Bouligand stationary point is characterized explicitly. The relationships among different stationary points under certain constraint qualifications are discussed. Then we propose a second order sufficient condition which can be weakened under the strict complementarity condition. Importantly, for the sake of algorithm design, under the assumption of strict complementarity condition, we give a nonsmooth equation reformulation of the stationary point, whose smoothing system is verified to be nonsingular at the stationary point under the proposed second order sufficient condition.