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On the linear independence constraint qualification in disjunctive programming

OPTIMIZATION 2020年第10期69卷 2241-2277页

作者： Mehlitz, Patrick Brandenburg Tech Univ Cottbus Senftenberg Inst Math Cottbus Germany

Mathematical programmes with disjunctive constraints (MPDCs for short) cover several different problem classes from nonlinear optimization including complementarity-, vanishing-, cardinality- and switching-constrained optimization problems. In this paper, we introduce an abstract but reasonable version of the prominent linear independence constraint qualification which applies to MPDCs. Afterwards, we derive first- and second-order optimality conditions for MPDCs under validity of this constraint qualification based on so-called strongly stationary points. Finally, we apply our findings to some popular classes of disjunctive programmes and compare the obtained results to those ones available in the literature. Particularly, new second-order optimality conditions for mathematical programmes with switching constraints are by-products of our approach.

关键词： constraint qualifications disjunctive programming linear independence constraint qualification strong stationarity second-order optimality conditions

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Convergence rate estimates for penalty methods revisited

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2023年第3期85卷 973-992页

作者： Izmailov, A. F. Solodov, M. V. Lomonosov Moscow State Univ VMK Fac OR Dept MSU Uchebniy Korpus 2 Moscow 119991 Russia IMPA Inst Matemat Pura & Aplicada Estr Dona Castorina 110 BR-22460320 Rio De Janeiro RJ Brazil

For the classical quadratic penalty, it is known that the distance from the solution of the penalty subproblem to the solution of the original problem is at worst inversely proportional to the value of the penalty parameter under the linear independence constraint qualification, strict complementarity, and the second-order sufficient optimality conditions. Moreover, using solutions of the penalty subproblem, one can obtain certain useful Lagrange multipliers estimates whose distance to the optimal ones is also at least inversely proportional to the value of the parameter. We show that the same properties hold more generally, namely, under the (weaker) strict Mangasarian-Fromovitz constraint qualification and second-order sufficiency (and without strict complementarity). Moreover, under the linear independence constraint qualification and strong second-order sufficiency (also without strict complementarity), we demonstrate local uniqueness and Lipschitz continuity of stationary points of penalty subproblems. In addition, those results follow from the analysis of general power penalty functions, of which quadratic penalty is a special case.

关键词： Penalty function Quadratic penalty Convergence rate Strong regularity linear independence constraint qualification Strict Mangasarian-Fromovitz constraint qualification Second-order sufficient optimality conditions

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Topological Approach to Mathematical Programs with Switching constraints

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SET-VALUED AND VARIATIONAL ANALYSIS 2022年第2期30卷 335-354页

作者： Shikhman, Vladimir Tech Univ Chemnitz Dept Math Reichenhainer Str 41 D-09126 Chemnitz Germany

We study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian's Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy and strong stability of W-stationary points. A complete characterization of strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, no bi-active Lagrange multipliers of a strongly stable W-stationary point can vanish.

关键词： Switching constraints W-stationarity W-index Morse theory Strong stability linear independence constraint qualification

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Certifiable relative pose estimation

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IMAGE AND VISION COMPUTING 2021年 109卷 104142-104142页

作者： Garcia-Salguero, Mercedes Briales, Jesus Gonzalez-Jimenez, Javier Univ Malaga Syst Engn & Automat Dept Machine Percept & Intelligent Robot MAPIR Grp Campus Teatinos Malaga 29071 Spain

In this paper we present the first fast optimality certifier for the non-minimal version of the Relative Pose prob-lem for calibrated cameras from epipolar constraints. The proposed certifier is based on Lagrangian duality and relies on a novel closed-form expression for dual points. We also leverage an efficient solver that performs local optimization on the manifold of the original problem's non-convex domain. The optimality of the solution is then checked via our novel fast certifier. The extensive conducted experiments demonstrate that, despite its simplicity, this certifiable solver performs excellently on synthetic data, repeatedly attaining the (certified a posteriori) optimal solution and shows a satisfactory performance on real data. (c) 2021 Elsevier B.V. All rights reserved.

关键词： Relative pose Essential matrix Epipolar constraint Convex programming Certifiable algorithm linear independence constraint qualification

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On the Sensitivity of a Euclidean Projection

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2014年第3期54卷 407-417页

作者： Izmailov, A. F. Kurennoy, A. S. Moscow MV Lomonosov State Univ Fac Computat Math & Cybernet Moscow 119992 Russia

The structure and behavior of Euclidean projections of a point onto a set defined by parametric constraints is studied. Under the Mangasarian-Fromovitz constraint qualification, it is shown that the projection is locally unique and continuous and, if the feasible set is constant, locally Lipschitz continuous as well. Quantitative results are obtained characterizing the asymptotic behavior of projections under perturbations in a given direction.

关键词： Euclidean projection sensitivity strong regularity strong stability Mangasarian-Fromovitz constraint qualification linear independence constraint qualification constant rank constraint qualification directional regularity

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Characterization of strong stability for C-stationary points in MPCC

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MATHEMATICAL PROGRAMMING 2012年第1-2期132卷 295-308页

作者： Jongen, H. Th. Shikhman, V. Steffensen, S. Rhein Westfal TH Aachen Dept Math C D-52056 Aachen Germany

We study mathematical programs with complementarity constraints (MPCC). Special focus will be on C-stationary points. Under the linear independence constraint qualification we characterize strong stability of C-stationary points (in the sense of Kojima) by means of first and second order information of the defining functions. It turns out that strong stability of C-stationary points allows a possible degeneracy of bi-active Lagrange multipliers. Some relations to other stationarity concepts (such as A-, M-, S- and B-stationarity) are shortly discussed.

关键词： Mathematical programs with complementarity constraints C-stationarity Strong stability linear independence constraint qualification

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MPCC: CRITICAL POINT THEORY

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SIAM JOURNAL ON OPTIMIZATION 2009年第1期20卷 473-484页

作者： Jongen, H. Th. Rueckmann, Jan. -J. Shikhman, V. Rhein Westfal TH Aachen Dept Math C D-52056 Aachen Germany Univ Birmingham Sch Math Birmingham B15 2TT W Midlands England

We study mathematical programs with complementarity constraints (MPCC) from a topological point of view. Special focus will be on C-stationary points. Under the linear independence constraint qualification (LICQ) we derive an equivariant Morse lemma at nondegenerate C-stationary points. Then, two basic theorems from Morse theory (deformation theorem and cell-attachment theorem) are proved. Outside the C-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a C-stationary level, the topology of the lower level set changes via the attachment of a q-dimensional cell. The dimension q equals the stationary C-index of the (nondegenerate) C-stationary point. The stationary C-index depends on both the restricted Hessian of the Lagrangian and the Lagrange multipliers related to biactive complementarity constraints. Finally, some relations with other stationarity concepts, such as W-, A-, M-, S-, and B-stationarity, are discussed.

关键词： mathematical programs with complementarity constraints C-stationarity linear independence constraint qualification stationary C-index Morse theory

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A sequential smooth penalization approach to mathematical programs with complementarity constraints

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2006年第1期27卷 71-98页

作者： Huang, XX Yang, XQ Zhu, DL Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China Chongqing Normal Univ Dept Math & Comp Sci Chongqing Peoples R China Fudan Univ Sch Management Shanghai 200433 Peoples R China

In this paper, a mathematical program with complementarity constraints (MPCC) is reformulated as a nonsmooth constrained mathematical program via the Fischer-Burmeister function. Smooth penalty functions are used to treat this nonsmooth constrained program. Under linear independence constraint qualification, and upper level strict complementarity condition, together with some other mild conditions, we prove that the limit point of stationary points satisfying second-order necessary conditions of unconstrained penalized problems is a strongly stationary point, hence a B-stationary point of the original MPCC. Furthermore, this limit point also satisfies a second-order necessary condition of the original MPCC. Numerical results are presented to test the performance of this method.

关键词： B-stationary point complementarity constraints linear independence constraint qualification mathematical program optimality condition penalty function

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Convergence of a penalty method for mathematical programming with complementarity constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2004年第2期123卷 365-390页

作者： Hu, XM Ralph, D CSIRO Mfg & Infrastruct Technol Sydney NSW Australia Univ Cambridge Cambridge England

We adapt the convergence analysis of the smoothing (Ref. 1) and regularization ( Ref. 2) methods to a penalty framework for mathematical programs with complementarity constraints (MPCC);we show that the penalty framework shares convergence properties similar to those of these methods. Moreover, we give sufficient conditions for a sequence generated by the penalty framework to be attracted to a B-stationary point of the MPCC.

关键词： complementarity constraints penalty methods B-stationarity linear independence constraint qualification

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New relaxation method for mathematical programs with complementarity constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2003年第1期118卷 81-116页

作者： Lin, GH Fukushima, M Kyoto Univ Dept Appl Math & Phys Grad Sch Informat Kyoto 606 Japan Dalian Univ Technol Dept Math Appl Dalian Peoples R China

In this paper, we present a new relaxation method for mathematical programs with complementarity constraints. Based on the fact that a variational inequality problem defined on a simplex can be represented by a finite number of inequalities, we use an expansive simplex instead of the nonnegative orthant involved in the complementarity constraints. We then remove some inequalities and obtain a standard nonlinear program. We show that the linear independence constraint qualification or the Mangasarian - Fromovitz constraint qualification holds for the relaxed problem under some mild conditions. We consider also a limiting behavior of the relaxed problem. We prove that any accumulation point of stationary points of the relaxed problems is a weakly stationary point of the original problem and that, if the function involved in the complementarity constraints does not vanish at this point, it is C-stationary. We obtain also some sufficient conditions of B-stationarity for a feasible point of the original problem. In particular, some conditions described by the eigenvalues of the Hessian matrices of the Lagrangian functions of the relaxed problems are new and can be verified easily. Our limited numerical experience indicates that the proposed approach is promising.

关键词： mathematical programs with equilibrium constraints linear independence constraint qualification nondegeneracy weak stationarity B-stationarity C-stationarity M-stationarity second-order necessary conditions upper level strict complementarity

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