Semidefinite relaxations for non-convex quadratic mixed-integer programming

We present semidefinite relaxations for unconstrained non-convex quadratic mixed-integer optimization problems. These relaxations yield tight bounds and are computationally easy to solve for medium-sized instances, even if some of the variables are integer and unbounded. In this case, the problem contains an infinite number of linear constraints; these constraints are separated dynamically. We use this approach as a bounding routine in an SDP-based branch-and-bound framework. In case of a convex objective function, the new SDP bound improves the bound given by the continuous relaxation of the problem. Numerical experiments show that our algorithm performs well on various types of non-convex instances.     

Gap inequalities for non-convex mixed-integer quadratic programs

Laurent and Poljak introduced a very general class of valid linear inequalities, called gap inequalities, for the max-cut problem. We show that an analogous class of inequalities can be defined for general non-convex mixed-integer quadratic programs. These inequalities dominate some inequalities arising from a natural semidefinite relaxation.     

Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations

We propose a deterministic global optimization approach, whose novel contributions are rooted in the edge-concave and piecewise-linear underestimators, to address nonconvex mixed-integer quadratically-constrained quadratic programs (MIQCQP) to ${\epsilon}$ -global optimality. The facets of low-dimensional (n ?? 3) edge-concave aggregations dominating the termwise relaxation of MIQCQP are introduced at every node of a branch-and-bound tree. Concave multivariable terms and sparsely distributed bilinear terms that do not participate in connected edge-concave aggregations are addressed through piecewise-linear relaxations. Extensive computational studies are presented for point packing problems, standard and generalized pooling problems, and examples from GLOBALLib (Meeraus, Globallib. http://www.gamsworld.org/global/globallib.htm).     

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.     

Convex dual for quadratic concave fractional programs

For a fractional program with a quadratic numerator and an arbitrary concave denominator, a new convex dual program is derived. Concepts of conjugate duality are used to obtain an explicit representation of the dual.The authors are grateful to two anonymous referees and the Associate Editor for their comments.     

Continuity of parametric mixed-integer quadratic programs and its application to stability analysis of two-stage quadratic stochastic programs with mixed-integer recourse

For mixed-integer quadratic program where all coefficients in the objective function and the right-hand sides of constraints vary simultaneously, we show locally Lipschitz continuity of its optimal value function, and derive the corresponding global estimation; furthermore, we also obtain quantitative estimation about the change of its optimal solutions. Applying these results to two-stage quadratic stochastic program with mixed-integer recourse, we establish quantitative stability of the optimal value function and the optimal solution set with respect to the Fortet-Mourier probability metric, when the underlying probability distribution is perturbed. The obtained results generalize available results on continuity properties of mixed-integer quadratic programs and extend current results on quantitative stability of two-stage quadratic stochastic programs with mixed-integer recourse.     

Quantitative stability of mixed-integer two-stage quadratic stochastic programs

For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is perturbed with respect to an appropriate probability metric. To this end, we first establish various Lipschitz continuity results about the value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of linear constraints. The obtained results extend earlier results about quantitative stability properties of stochastic integer programming and stability results for mixed-integer parametric quadratic programs.     

Improved algorithm for mixed-integer quadratic programs and a computational study

In a recent paper [6] we suggested an algorithm for solving complicated mixed-integer quadratic programs, based on an equivalent formulation that employs a nonsingular transformation of variables. The objectives of the present paper are two. First, to present an improved version of this algorithm, which reduces substantially its computational requirements; second, to report on the results of a computational study with the revised algorithm.     

Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs

A rigorous decomposition approach to solve separable mixed-integer nonlinear programs where the participating functions are nonconvex is presented. The proposed algorithms consist of solving an alternating sequence of Relaxed Master Problems (mixed-integer linear program) and two nonlinear programming problems (NLPs). A sequence of valid nondecreasing lower bounds and upper bounds is generated by the algorithms which converge in a finite number of iterations. A Primal Bounding Problem is introduced, which is a convex NLP solved at each iteration to derive valid outer approximations of the nonconvex functions in the continuous space. Two decomposition algorithms are presented in this work. On finite termination, the first yields the global solution to the original nonconvex MINLP and the second finds a rigorous bound to the global solution. Convergence and optimality properties, and refinement of the algorithms for efficient implementation are presented. Finally, numerical results are compared with currently available algorithms for example problems, illuminating the potential benefits of the proposed algorithm.     

Dual formulations and subgradient optimization strategies for linear programming relaxations of mixed-integer programs

This paper examines algorithmic strategies relating to the formulation of Lagrangian duals, and their solution via subgradient optimization, in the context of solving linear programming relaxations of mixed-integer programs. As in the current trend in integer programming, several researchers have found it beneficial to generate and add additional constraints to the model, prior to its solution, in order to tighten its linear programming relaxation. It becomes necessary, therefore, to be able to efficiently derive strong surrogate constraints and bounds based on such reformulations. This paper addresses the design and testing of the most viable technique available for exploiting such tight reformulations, namely, using Lagrangian relaxation in coordination with subgradient optimization. A computational study is conducted to test the efficiency of various Lagrangian dual formulations, and to investigate in the context of using subgradient optimization the effects of step size choices, problem scaling, conducting pattern moves, and projecting dual solutions onto subsets of violated dual constraints. Based on this study, certain recommendations are made regarding the manner in which one should implement such an approach.     

Global solution of nonlinear mixed-integer bilevel programs

An algorithm for the global optimization of nonlinear bilevel mixed-integer programs is presented, based on a recent proposal for continuous bilevel programs by Mitsos et al. (J Glob Optim 42(4):475–513, 2008). The algorithm relies on a convergent lower bound and an optional upper bound. No branching is required or performed. The lower bound is obtained by solving a mixed-integer nonlinear program, containing the constraints of the lower-level and upper-level programs; its convergence is achieved by also including a parametric upper bound to the optimal solution function of the lower-level program. This lower-level parametric upper bound is based on Slater-points of the lower-level program and subsets of the upper-level host sets for which this point remains lower-level feasible. Under suitable assumptions the KKT necessary conditions of the lower-level program can be used to tighten the lower bounding problem. The optional upper bound to the optimal solution of the bilevel program is obtained by solving an augmented upper-level problem for fixed upper-level variables. A convergence proof is given along with illustrative examples. An implementation is described and applied to a test set comprising original and literature problems. The main complication relative to the continuous case is the construction of the parametric upper bound to the lower-level optimal objective value, in particular due to the presence of upper-level integer variables. This challenge is resolved by performing interval analysis over the convex hull of the upper-level integer variables.     

Branch-and-price for a class of nonconvex mixed-integer nonlinear programs

This work attempts to combine the strengths of two major technologies that have matured over the last three decades: global mixed-integer nonlinear optimization and branch-and-price. We consider a class of generally nonconvex mixed-integer nonlinear programs (MINLPs) with linear complicating constraints and integer linking variables. If the complicating constraints are removed, the problem becomes easy to solve, e.g. due to decomposable structure. Integrality of the linking variables allows us to apply a discretization approach to derive a Dantzig-Wolfe reformulation and solve the problem to global optimality using branch-andprice. It is a remarkably simple idea; but to our surprise, it has barely found any application in the literature. In this work, we show that many relevant problems directly fall or can be reformulated into this class of MINLPs. We present the branch-and-price algorithm and demonstrate its effectiveness (and sometimes ineffectiveness) in an extensive computational study considering multiple large-scale problems of practical relevance, showing that, in many cases, orders-of-magnitude reductions in solution time can be achieved.

     

Faster,but weaker,relaxations for quadratically constrained quadratic programs

We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed block-diagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.     

A note on set-semidefinite relaxations of nonconvex quadratic programs

We consider semidefinite, copositive, and more general, set-semidefinite programming relaxations of general nonconvex quadratic problems. For the semidefinite case a comparison between the feasible set of the original program and the feasible set of the relaxation has been given by Kojima and Tunçel (SIAM J Optim 10(3):750–778, 2000). In this paper the comparison is presented for set-positive relaxations which contain copositive relaxations as a special case.     

Linearization-based algorithms for mixed-integer nonlinear programs with convex continuous relaxation

We present two linearization-based algorithms for mixed-integer nonlinear programs (MINLPs) having a convex continuous relaxation. The key feature of these algorithms is that, in contrast to most existing linearization-based algorithms for convex MINLPs, they do not require the continuous relaxation to be defined by convex nonlinear functions. For example, these algorithms can solve to global optimality MINLPs with constraints defined by quasiconvex functions. The first algorithm is a slightly modified version of the LP/NLP-based branch-and-bouund \((\text{ LP/NLP-BB })\) algorithm of Quesada and Grossmann, and is closely related to an algorithm recently proposed by Bonami et al. (Math Program 119:331–352, 2009). The second algorithm is a hybrid between this algorithm and nonlinear programming based branch-and-bound. Computational experiments indicate that the modified LP/NLP-BB method has comparable performance to LP/NLP-BB on instances defined by convex functions. Thus, this algorithm has the potential to solve a wider class of MINLP instances without sacrificing performance.     

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

An outer-approximation algorithm is presented for solving mixed-integer nonlinear programming problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the proposed algorithm effectively exploits the structure of the problems, and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed-integer linear master program. Convergence and optimality properties of the algorithm are presented, as well as a general discussion on its implementation. Numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper. Finally, a theoretical comparison with generalized Benders decomposition is presented on the lower bounds predicted by the relaxed master programs.     

Strong relaxations for continuous nonlinear programs based on decision diagrams

Decision Diagrams (DDs) have arisen as a powerful tool to solve discrete optimization problems. The extension of this emerging concept to continuous problems, however, has remained a challenge. In this paper, we introduce a novel framework that utilizes DDs to model continuous nonlinear programs. This framework, when combined with the array of techniques available for discrete problems, illuminates a new pathway to solving mixed integer nonlinear programs with the help of DDs.     

Foundation-penalty cuts for mixed-integer programs

We propose a new class of foundation-penalty (FP) cuts for MIPs that are easy to generate by exploiting routine penalty calculations. Their underlying concept generalizes the lifting process and provides derivations of major classical cuts. (Gomory cuts arise from low level FP cuts by simply ‘plugging in’ standard penalties.)     

On standard quadratic programs with exact and inexact doubly nonnegative relaxations

Mathematical Programming - The problem of minimizing a (nonconvex) quadratic form over the unit simplex, referred to as a standard quadratic program, admits an exact convex conic formulation over...