Xiaojin Zheng

Improving the performance of MIQP solvers for quadratic programs with cardinality and minimum threshold constraints: A semidefinite program approach

We consider in this paper quadratic programming problems with cardinality and minimum threshold constraints that arise naturally in various real-world applications such as portfolio selection and subset selection in regression. This class of problems can be formulated as mixed-integer 0-1 quadratic programs. We propose a new semidefinite program (SDP) approach for computing the “best” diagonal decomposition that gives the tightest continuous relaxation of the perspective reformulation of the problem. We also give an alternative way of deriving the perspective reformulation by applying a special Lagrangian decomposition scheme to the diagonal decomposition of the problem. This derivation can be viewed as a “dual” method to the convexification method employing the perspective function on semicontinuous variables. Computational results show that the proposed SDP approach can be advantageous for improving the performance of mixed-integer quadratic programming solvers when applied to the perspective reformulations of the problem.

Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach

In this paper we consider cardinality-constrained convex programs that minimize a convex function subject to a cardinality constraint and other linear constraints. This class of problems has found many applications, including portfolio selection, subset selection and compressed sensing. We propose a successive convex approximation method for this class of problems in which the cardinality function is first approximated by a piecewise linear DC function (difference of two convex functions) and a sequence of convex subproblems is then constructed by successively linearizing the concave terms of the DC function. Under some mild assumptions, we establish that any accumulation point of the sequence generated by the method is a KKT point of the DC approximation problem. We show that the basic algorithm can be refined by adding strengthening cuts in the subproblems. Finally, we report some preliminary computational results on cardinality-constrained portfolio selection problems.

Lagrangian decomposition and mixed-integer quadratic programming reformulations for probabilistically constrained quadratic programs

Probabilistically constrained quadratic programming (PCQP) problems arise naturally from many real-world applications and have posed a great challenge in front of the optimization society for years due to the nonconvex and discrete nature of its feasible set. We consider in this paper a special case of PCQP where the random vector has a finite discrete distribution. We first derive second-order cone programming (SOCP) relaxation and semidefinite programming (SDP) relaxation for the problem via a new Lagrangian decomposition scheme. We then give a mixed integer quadratic programming (MIQP) reformulation of the PCQP and show that the continuous relaxation of the MIQP is exactly the SOCP relaxation. This new MIQP reformulation is more efficient than the standard MIQP reformulation in the sense that its continuous relaxation is tighter than or at least as tight as that of the standard MIQP. We report preliminary computational results to demonstrate the tightness of the new convex relaxations and the effectiveness of the new MIQP reformulation.

Duality Gap Estimation of Linear Equality Constrained Binary Quadratic Programming

We investigate in this paper the Lagrangian duality properties of linear equality constrained binary quadratic programming. We derive an underestimation of the duality gap between the primal problem and its Lagrangian dual or SDP relaxation, using the distance from the set of binary integer points to certain affine subspace, while the computation of this distance can be achieved by the cell enumeration of hyperplane arrangement. Alternative Lagrangian dual schemes via the exact penalty and the squared norm constraint reformulations are also discussed.

Reachability determination in acyclic Petri nets by cell enumeration approach

Reachability is one of the most important behavioral properties of Petri nets. We propose in this paper a novel approach for solving the fundamental equation in the reachability analysis of acyclic Petri nets, which has been known to be NP-complete. More specifically, by adopting a revised version of the cell enumeration method for an arrangement of hyperplanes in discrete geometry, we develop an efficient solution scheme to identify firing count vector solution(s) to the fundamental equation on a bounded integer set, with a complexity bound of O ( ( n u ) n - m ) , where n is the number of transitions, m is the number of places and u is the upper bound of the number of firings for all individual transitions.

ON THE REDUCTION OF DUALITY GAP IN BOX CONSTRAINED NONCONVEX QUADRATIC PROGRAM

In this paper, we investigate in this paper the reduction of the duality gap between box constrained nonconvex quadratic programming and its semidefinite programming (SDP) relaxation (or Lagrangian dual). Characterizing the zero duality gap by a set of saddle-point-type conditions, we propose a parameterized distance measure δ(θ) between a polyhedral set C and a perturbed nonconvex set Λ(θ) to measure the dissatisfaction degree of the optimality conditions for zero duality gap. An underestimation of the duality gap is then derived which leads to a reduction of the duality gap proportional to δ2(θ*) for the identified best parameter θ*. This reduction of duality gap can be extended to the cases with both box and linear equality constraints. We demonstrate that the computation of δ(θ*) can be reduced to the cell enumeration of hyperplane arrangement in discrete geometry. In particular, we show that the reduction of duality gap can be achieved in polynomial time for a fixed degeneracy degree of the modified ...

Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

We consider in this paper a class of semi-continuous quadratic programming problems which arises in many real-world applications such as production planning, portfolio selection and subset selection in regression. We propose a lift-and-convexification approach to derive an equivalent reformulation of the original problem. This lift-and-convexification approach lifts the quadratic term involving

Cell-and-bound algorithm for chance constrained programs with discrete distributions

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A note on semidefinite relaxation for 0-1 quadratic knapsack problems

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Recent Advances in Mathematical Programming with Semi-continuous Variables and Cardinality Constraint

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