Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming

Decomposition has proved to be one of the more effective tools for the solution of large-scale problems, especially those arising in stochastic programming. A decomposition method with wide applicability is Benders' decomposition, which has been applied to both stochastic programming as well as integer programming problems. However, this method of decomposition relies on convexity of the value function of linear programming subproblems. This paper is devoted to a class of problems in which the second-stage subproblem(s) may impose integer restrictions on some variables. The value function of such integer subproblem(s) is not convex, and new approaches must be designed. In this paper, we discuss alternative decomposition methods in which the second-stage integer subproblems are solved using branch-and-cut methods. One of the main advantages of our decomposition scheme is that Stochastic Mixed-Integer Programming (SMIP) problems can be solved by dividing a large problem into smaller MIP subproblems that can be solved in parallel. This paper lays the foundation for such decomposition methods for two-stage stochastic mixed-integer programs.     

Column enumeration based decomposition techniques for a class of non-convex MINLP problems

We propose a decomposition algorithm for a special class of nonconvex mixed integer nonlinear programming problems which have an assignment constraint. If the assignment decisions are decoupled from the remaining constraints of the optimization problem, we propose to use a column enumeration approach. The master problem is a partitioning problem whose objective function coefficients are computed via subproblems. These problems can be linear, mixed integer linear, (non-)convex nonlinear, or mixed integer nonlinear. However, the important property of the subproblems is that we can compute their exact global optimum quickly. The proposed technique will be illustrated solving a cutting problem with optimum nonlinear programming subproblems.     

Equivalence of some quadratic programming algorithms

We formulate a general algorithm for the solution of a convex (but not strictly convex) quadratic programming problem. Conditions are given under which the iterates of the algorithm are uniquely determined. The quadratic programming algorithms of Fletcher, Gill and Murray, Best and Ritter, and van de Panne and Whinston/Dantzig are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points. The various methods are shown to differ only in the manner in which they solve the linear equations expressing the Kuhn-Tucker system for the associated equality constrained subproblems. Equivalence results have been established by Goldfarb and Djang for the positive definite Hessian case. Our analysis extends these results to the positive semi-definite case. This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A8189.     

Maximization of the Ratio of Two Convex Quadratic Functions over a Polytope

In this paper, we will develop an algorithm for solving a quadratic fractional programming problem which was recently introduced by Lo and MacKinlay to construct a maximal predictability portfolio, a new approach in portfolio analysis. The objective function of this problem is defined by the ratio of two convex quadratic functions, which is a typical global optimization problem with multiple local optima. We will show that a well-designed branch-and-bound algorithm using (i) Dinkelbach's parametric strategy, (ii) linear overestimating function and (iii) -subdivision strategy can solve problems of practical size in an efficient way. This algorithm is particularly efficient for Lo-MacKinlay's problem where the associated nonconvex quadratic programming problem has low rank nonconcave property.     

A linear programming-based optimization algorithm for solving nonlinear programming problems

In this paper a linear programming-based optimization algorithm called the Sequential Cutting Plane algorithm is presented. The main features of the algorithm are described, convergence to a Karush–Kuhn–Tucker stationary point is proved and numerical experience on some well-known test sets is showed. The algorithm is based on an earlier version for convex inequality constrained problems, but here the algorithm is extended to general continuously differentiable nonlinear programming problems containing both nonlinear inequality and equality constraints. A comparison with some existing solvers shows that the algorithm is competitive with these solvers. Thus, this new method based on solving linear programming subproblems is a good alternative method for solving nonlinear programming problems efficiently. The algorithm has been used as a subsolver in a mixed integer nonlinear programming algorithm where the linear problems provide lower bounds on the optimal solutions of the nonlinear programming subproblems in the branch and bound tree for convex, inequality constrained problems.     

The mixed vertex packing problem

We study a generalization of the vertex packing problem having both binary and bounded continuous variables, called the mixed vertex packing problem (MVPP). The well-known vertex packing model arises as a subproblem or relaxation of many 0-1 integer problems, whereas the mixed vertex packing model arises as a natural counterpart of vertex packing in the context of mixed 0-1 integer programming. We describe strong valid inequalities for the convex hull of solutions to the MVPP and separation algorithms for these inequalities. We give a summary of computational results with a branch-and-cut algorithm for solving the MVPP and using it to solve general mixed-integer problems. Received: June 1998 / Accepted: February 2000?Published online September 20, 2000     

A simplicial branch and duality bound algorithm for the sum of convex–convex ratios problem

This article presents a simplicial branch and duality bound algorithm for globally solving the sum of convex–convex ratios problem with nonconvex feasible region. To our knowledge, little progress has been made for globally solving this problem so far. The algorithm uses a branch and bound scheme where the Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems during the algorithm search are all ordinary linear programs that can be solved very efficiently. It has been proved that the algorithm possesses global convergence. Finally, the numerical experiments are given to show the feasibility of the proposed algorithm.     

一类带反凸约束的非线性比式和问题的全局优化算法

本文针对一类带有反凸约束的非线性比式和分式规划问题,提出一种求其全局最优解的单纯形分支和对偶定界算法.该算法利用Lagrange对偶理论将其中关键的定界问题转化为一系列易于求解的线性规划问题.收敛性分析和数值算例均表明提出的算法是可行的.     

Exact Algorithm for the Surrogate Dual of an Integer Programming Problem: Subgradient Method Approach

One of the critical issues in the effective use of surrogate relaxation for an integer programming problem is how to solve the surrogate dual within a reasonable amount of computational time. In this paper, we present an exact and efficient algorithm for solving the surrogate dual of an integer programming problem. Our algorithm follows the approach which Sarin et al. (Ref. 8) introduced in their surrogate dual multiplier search algorithms. The algorithms of Sarin et al. adopt an ad-hoc stopping rule in solving subproblems and cannot guarantee the optimality of the solutions obtained. Our work shows that this heuristic nature can actually be eliminated. Convergence proof for our algorithm is provided. Computational results show the practical applicability of our algorithm.     

A simplex social spider algorithm for solving integer programming and minimax problems

In this paper, we propose a new hybrid social spider algorithm with simplex Nelder-Mead method in order to solve integer programming and minimax problems. We call the proposed algorithm a Simplex Social Spider optimization (SSSO) algorithm. In the the proposed SSSO algorithm, we combine the social spider algorithm with its powerful capability of performing exploration, exploitation, and the Nelder-Mead method in order to refine the best obtained solution from the standard social spider algorithm. In order to investigate the general performance of the proposed SSSO algorithm, we test it on 7 integer programming problems and 10 minimax problems and compare against 10 algorithms for solving integer programming problems and 9 algorithms for solving minimax problems. The experiments results show the efficiency of the proposed algorithm and its ability to solve integer and minimax optimization problems in reasonable time.     

Sequential Systems of Linear Equations Algorithm for Nonlinear Optimization Problems with General Constraints

In Ref. 1, a new superlinearly convergent algorithm of sequential systems of linear equations (SSLE) for nonlinear optimization problems with inequality constraints was proposed. At each iteration, this new algorithm only needs to solve four systems of linear equations having the same coefficient matrix, which is much less than the amount of computation required for existing SQP algorithms. Moreover, unlike the quadratic programming subproblems of the SQP algorithms (which may not have a solution), the subproblems of the SSLE algorithm are always solvable. In Ref. 2, it is shown that the new algorithm can also be used to deal with nonlinear optimization problems having both equality and inequality constraints, by solving an auxiliary problem. But the algorithm of Ref. 2 has to perform a pivoting operation to adjust the penalty parameter per iteration. In this paper, we improve the work of Ref. 2 and present a new algorithm of sequential systems of linear equations for general nonlinear optimization problems. This new algorithm preserves the advantages of the SSLE algorithms, while at the same time overcoming the aforementioned shortcomings. Some numerical results are also reported.     

Comparison of two algorithms for solving a two‐stage bilinear stochastic programming problem with quantile criterion

The paper is devoted to solving the two‐stage problem of stochastic programming with quantile criterion. It is assumed that the loss function is bilinear in random parameters and strategies, and the random vector has a normal distribution. Two algorithms are suggested to solve the problem, and they are compared. The first algorithm is based on the reduction of the original stochastic problem to a mixed integer linear programming problem. The second algorithm is based on the reduction of the problem to a sequence of convex programming problems. Performance characteristics of both the algorithms are illustrated by an example. A modification of both the algorithms is suggested to reduce the computing time. The new algorithm uses the solution obtained by the second algorithm as a starting point for the first algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

Outer-approximation algorithms for nonsmooth convex MINLP problems

In this work, we combine outer-approximation (OA) and bundle method algorithms for dealing with mixed-integer non-linear programming (MINLP) problems with nonsmooth convex objective and constraint functions. As the convergence analysis of OA methods relies strongly on the differentiability of the involved functions, OA algorithms may fail to solve general nonsmooth convex MINLP problems. In order to obtain OA algorithms that are convergent regardless the structure of the convex functions, we solve the underlying OA’s non-linear subproblems by a specialized bundle method that provides necessary information to cut off previously visited (non-optimal) integer points. This property is crucial for proving (finite) convergence of OA algorithms. We illustrate the numerical performance of the given proposal on a class of hybrid robust and chance-constrained problems that involve a random variable with finite support.     

Plant location with minimum inventory

We present an integer programming model for plant location with inventory costs. The linear programming relaxation has been solved by Dantzig-Wolfe decomposition. In this case the subproblems reduce to the minimum cut problem. We have used subgradient optimization to accelerate the convergence of the D-W algorithm. We present our experience with problems arising in the design of a distribution network for computer spare parts. In most cases, from a fractional solution we were able to derive integer solutions within 4% of optimality.     

A sequential cutting plane algorithm for solving convex NLP problems

In this paper we look at a new algorithm for solving convex nonlinear programming optimization problems. The algorithm is a cutting plane-based method, where the sizes of the subproblems remain fixed, thus avoiding the issue with constantly growing subproblems we have for the classical Kelley’s cutting plane algorithm. Initial numerical experiments indicate that the algorithm is considerably faster than Kelley’s cutting plane algorithm and also competitive with existing nonlinear programming algorithms.     

SEQUENTIAL CONVEX PROGRAMMING METHODS FOR SOLVING LARGE TOPOLOGY OPTIMIZATION PROBLEMS： IMPLEMENTATION AND COMPUTATIONAL RESULTS

In this paper, we describe a method to solve large-scale structural optimization problems by sequential convex programming （SCP）. A predictor-corrector interior point method is applied to solve the strictly convex subproblems. The SCP algorithm and the topology optimization approach are introduced. Especially, different strategies to solve certain linear systems of equations are analyzed. Numerical results are presented to show the efficiency of the proposed method for solving topology optimization problems and to compare different variants.     

A sequential equality constrained quadratic programming algorithm for inequality constrained optimization

In this paper, the feasible type SQP method is improved. A new SQP algorithm is presented to solve the nonlinear inequality constrained optimization. As compared with the existing SQP methods, per single iteration, in order to obtain the search direction, it is only necessary to solve equality constrained quadratic programming subproblems and systems of linear equations. Under some suitable conditions, the global and superlinear convergence can be induced.     

Branch-and-Bound Outer Approximation Algorithm for Sum-of-Ratios Fractional Programs

In this article, a branch and-bound outer approximation algorithm is presented for globally solving a sum-of-ratios fractional programming problem. To solve this problem, the algorithm instead solves an equivalent problem that involves minimizing an indefinite quadratic function over a nonempty, compact convex set. This problem is globally solved by a branch-and-bound outer approximation approach that can create several closed-form linear inequality cuts per iteration. In contrast to pure outer approximation techniques, the algorithm does not require computing the new vertices that are created as these cuts are added. Computationally, the main work of the algorithm involves solving a sequence of convex programming problems whose feasible regions are identical to one another except for certain linear constraints. As a result, to solve these problems, an optimal solution to one problem can potentially be used to good effect as a starting solution for the next problem.     

The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming

A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in this paper: The extended supporting hyperplane (ESH) algorithm uses supporting hyperplanes to generate a tight overestimated polyhedral set of the feasible set defined by linear and nonlinear constraints. A sequence of linear or quadratic integer-relaxed subproblems are first solved to rapidly generate a tight linear relaxation of the original MINLP problem. After an initial overestimated set has been obtained the algorithm solves a sequence of mixed-integer linear programming or mixed-integer quadratic programming subproblems and refines the overestimated set by generating more supporting hyperplanes in each iteration. Compared to the extended cutting plane algorithm ESH generates a tighter overestimated set and unlike outer approximation the generation point for the supporting hyperplanes is found by a simple line search procedure. In this paper it is proven that the ESH algorithm converges to a global optimum for convex MINLP problems. The ESH algorithm is implemented as the supporting hyperplane optimization toolkit (SHOT) solver, and an extensive numerical comparison of its performance against other state-of-the-art MINLP solvers is presented.     

New convergent heuristics for 0–1 mixed integer programming

Several hybrid methods have recently been proposed for solving 0–1 mixed integer programming problems. Some of these methods are based on the complete exploration of small neighborhoods. In this paper, we present several convergent algorithms that solve a series of small sub-problems generated by exploiting information obtained from a series of relaxations. These algorithms generate a sequence of upper bounds and a sequence of lower bounds around the optimal value. First, the principle of a linear programming-based algorithm is summarized, and several enhancements of this algorithm are presented. Next, new hybrid heuristics that use linear programming and/or mixed integer programming relaxations are proposed. The mixed integer programming (MIP) relaxation diversifies the search process and introduces new constraints in the problem. This MIP relaxation also helps to reduce the gap between the final upper bound and lower bound. Our algorithms improved 14 best-known solutions from a set of 108 available and correlated instances of the 0–1 multidimensional Knapsack problem. Other encouraging results obtained for 0–1 MIP problems are also presented.