Cut Generation for Mixed 0-1 Quadratically Constrained Programs (original) (raw)

Lift-and-Project Cutting Planes for Mixed 0-1 Semidefinite Programming

This paper develops a cutting plane method for solving the mixed 0-1 semidefinite program (MSDP) minimize c T x; subject to F(x) = F 0 + P n i=1 x i F i 0; x i 2 f0; 1g; i = 1; : : : ; p; (1) where the decision variable x 2 R n and F i = F T i 2 R m m for i = 0; : : : ; n. We assume that the linear inequalities 0 x i 1 for 1 i p are already present in the semidefinite constraint F(x) 0. The ane semidefinite constraint F(x) 0 in (1) is called a linear matrix inequality (LMI). In the early 1990s, Nemirovski and Nesterov [34] developed interior point algorithms for solving self-dual homogeneous conic programs, which include semidefinite programs (SDPs), second-order cone programs (SOCPs) and linear programs (LPs) as special cases. Amongst these SDPs are the most general in the sense that LPs and SOCPs can be reformulated as SDPs. Conic constraints include a large variety convex constraints such as linear constraints, convex quadratic constraints, linear fractional constraints, eigenval...

An optimality cut for mixed integer linear programs

European Journal of Operational Research, 1999

We derive the penalty cut, a simple optimality cut of general applicability in pure or mixed linear programs. This cut is tested on a number of examples and comparisons with the classical Gomory cut are provided.

The method for second order cone programming

Computers & Operations Research, 2008

We develop the Q method for the second order cone programming problem. The algorithm is the adaptation of the Q method for semidefinite programming originally developed by Alizadeh, Haeberly and Overton [A new primal-dual interior point method for semidefinite programming. In: Proceedings of the fifth SIAM conference on applications of linear algebra, Snowbird, Utah, 1994.] and [Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results. SIAM Journal on Optimization 1998;8(3):746-68 [electronic].]. We take advantage of the special algebraic structure associated with second order cone programs to formulate the Q method. Furthermore we discuss the convergence properties of the algorithm. Finally, some numerical results are presented. ᭧

Generating Cutting Inequalities Successively for Quadratic Optimization Problems in Binary Variables

2021

We propose a successive generation of cutting inequalities for binary quadratic optimization problems. Multiple cutting inequalities are successively generated for the convex hull of the set of the optimal solutions ⊂ {0, 1}n, while the standard cutting inequalities are used for the convex hull of the feasible region. An arbitrary linear inequality with integer coefficients and the right-hand side value in integer is considered as a candidate for a valid inequality. The validity of the linear inequality is determined by solving a conic relaxation of a subproblem such as the doubly nonnegative relaxation, under the assumption that an upper bound for the unknown optimal value of the problem is available. Moreover, the candidates generated for the multiple cutting inequalities are tested simultaneously for their validity in parallel. Preliminary numerical results on 60 quadratic unconstrained binary optimization problems with a simple implementation of the successive cutting inequaliti...

BENDERS DECOMPOSITION FOR CONE PROGRAMMING

RUTCOR Research Report RRR16-1996, 1996

Cone programming is a joint generalization of the pozitive semi-definite programming and linear programming. The requirement that the matrix of variables must be pozitive definite or the vector of variables must be nonnegative resp., in semi-definite or linear programming, resp., is substituted by the claim that the vector of variables must belong to a fixed and not necessarily polyhedral cone. The original Benders decomposition is a frame of algorithms to solve optimization problems such that the variables can be divided into two sets in such a way that for every fixed values of the variables of the second set the reduced problem is a linear programming problem. In this paper Benders decomposition is generalized for the case when the reduced problem is a problem of cone programming.

Applications of Second-Order Cone Programming

Linear Algebra and Its Applications, 1998

In a second-Order cone program (SOCP) a linear function is minimized over the intersection of an affine set and the product of second-Order (quadratic) cones. SOCPs are nonlinear convex Problems that include linear and (convex) quadratic programs as special cases, but are less general than semidefinite programs (SDPs). Several efficient pri- O<cTx+di, i= l,..., N.

A branch-and-cut algorithm for nonconvex quadratic programs with box constraints

Mathematical Programming, 2005

We present a branch and cut algorithm that yields in finite time, a globally -optimal solution (with respect to feasibility and optimality) of the nonconvex quadratically constrained quadratic programming problem. The idea is to estimate all quadratic terms by successive linearizations within a branching tree using Reformulation-Linearization Techniques (RLT). To do so, four classes of linearizations (cuts), depending on one to three parameters, are detailed. For each class, we show how to select the best member with respect to a precise criterion. The cuts introduced at any node of the tree are valid in the whole tree, and not only within the subtree rooted at that node. In order to enhance the computational speed, the structure created at any node of the tree is flexible enough to be used at other nodes. Computational results are reported that include standard test problems taken from the literature. Some of these problems are solved for the first time with a proof of global optimality.