On linear programs with linear complementarity constraints

Hu, Jing; Mitchell, John E.; Pang, Jong-Shi; Yu, Bin

doi:10.1007/s10898-010-9644-3

Cited by 57 publications

(33 citation statements)

References 53 publications

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“…To test the validity of the constraint (21), we set z i = 0 for all i ∈ I 1 and z j = 1 for all j ∈ J 1 in (20) and restore the complementarity formulation of the resulting restricted IP:…”

Section: Specialization To Solvable Qpsmentioning

confidence: 99%

“…Presently, we are actively researching this general issue. The reader is referred to [32] for some bounding and enveloping techniques for handling products of variables, to [31] for a disjunctive approach to handling the quadratic equality (23), and to the Ph.D. dissertation [20] for the detailed specialization of the scheme to problems with bounded variables. Linear programming relaxations of (4) and (22) can be tightened by adding constraints based on second-order optimality conditions for the quadratic program (1), as we discuss in the remainder of this section.…”

Section: Specialization To Solvable Qpsmentioning

confidence: 99%

“…Proposition 3 motivates some valid inequalities for the MIP (20). To introduce these inequalities, we define the family of index sets: …”

Section: Proposition 3 If X Is a Local Minimum Of The Qp (1) Then Q mentioning

confidence: 99%

“…We call each inequality in Z 2 corresponding to a J ∈ J a 2nd-order cut of the mixed IP (20). If A has full column rank, then every index set J ∈ J must be a proper subset of {1, .…”

Section: Proposition 3 If X Is a Local Minimum Of The Qp (1) Then Q mentioning

confidence: 99%

“…We refer the reader to the Ph.D. dissertation [20] for the detailed specialization of the scheme, including discussion of the point and ray cuts and their sparsification.…”

Section: Simply Constrained Qpsmentioning

confidence: 99%

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An LPCC approach to nonconvex quadratic programs

Mitchell

Pang

2010

Math. Program.

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Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving two linear programs with linear complementarity constraints, i.e., LPCCs. Specifically, this task can be divided into two LPCCs: the first confirms whether the QP is bounded below on the feasible set and, if not, computes a feasible ray on which the QP is unbounded; the second LPCC computes a globally optimal solution if it exists, by identifying a stationary point that yields the best quadratic objective value. In turn, the global resolution of these LPCCs can be accomplished by a parameter-free, mixed integer-programming based, finitely terminating algorithm developed recently by the authors, which can be enhanced in this context by a new kind of valid cut derived from the second-order conditions of the QP and by exploiting the special structure of the LPCCs. Throughout, our treatment makes no boundedness assumption 123 244 J. Hu et al.of the QP; this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption. The general theory is illustrated by 3 classes of indefinite problems: QPs with simple upper and lower bounds (existence of optimal solutions is guaranteed); same QPs with an additional inequality constraint (extending the case of simple bound constraints); and nonnegatively constrained copositive QPs (no guarantee of the existence of an optimal solution). We also present numerical results to support the special cuts obtained due to the QP connection.

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Section: Specialization To Solvable Qpsmentioning

confidence: 99%

Section: Specialization To Solvable Qpsmentioning

confidence: 99%

“…Proposition 3 motivates some valid inequalities for the MIP (20). To introduce these inequalities, we define the family of index sets: …”

Section: Proposition 3 If X Is a Local Minimum Of The Qp (1) Then Q mentioning

confidence: 99%

“…We call each inequality in Z 2 corresponding to a J ∈ J a 2nd-order cut of the mixed IP (20). If A has full column rank, then every index set J ∈ J must be a proper subset of {1, .…”

Section: Proposition 3 If X Is a Local Minimum Of The Qp (1) Then Q mentioning

confidence: 99%

“…We refer the reader to the Ph.D. dissertation [20] for the detailed specialization of the scheme, including discussion of the point and ray cuts and their sparsification.…”

Section: Simply Constrained Qpsmentioning

confidence: 99%

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An LPCC approach to nonconvex quadratic programs

Mitchell

Pang

2010

Math. Program.

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Deterministic Consensus Maximization with Biconvex Programming

Cai

Chin

et al. 2018

Lecture Notes in Computer Science

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Consensus maximization is one of the most widely used robust fitting paradigms in computer vision, and the development of algorithms for consensus maximization is an active research topic. In this paper, we propose an efficient deterministic optimization algorithm for consensus maximization. Given an initial solution, our method conducts a deterministic search that forcibly increases the consensus of the initial solution. We show how each iteration of the update can be formulated as an instance of biconvex programming, which we solve efficiently using a novel biconvex optimization algorithm. In contrast to our algorithm, previous consensus improvement techniques rely on random sampling or relaxations of the objective function, which reduce their ability to significantly improve the initial consensus. In fact, on challenging instances, the previous techniques may even return a worse off solution. Comprehensive experiments show that our algorithm can consistently and greatly improve the quality of the initial solution, without substantial cost. 4

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