A comparative study of SQP-type algorithms for nonlinear and nonconvex mixed-integer optimization

We present numerical results of a comparative study of codes for nonlinear and nonconvex mixed-integer optimization. The underlying algorithms are based on sequential quadratic programming (SQP) with stabilization by trust-regions, linear outer approximations, and branch-and-bound techniques. The mixed-integer quadratic programming subproblems are solved by a branch-and-cut algorithm. Second order information is updated by a quasi-Newton update formula (BFGS) applied to the Lagrange function for continuous, but also for integer variables. We do not require that the model functions can be evaluated at fractional values of the integer variables. Thus, partial derivatives with respect to integer variables are replaced by descent directions obtained from function values at neighboring grid points, and the number of simulations or function evaluations, respectively, is our main performance criterion to measure the efficiency of a code. Numerical results are presented for a set of 100 academic mixed-integer test problems. Since not all of our test examples are convex, we reach the best-known solutions in about 90 % of the test runs, but at least feasible solutions in the other cases. The average number of function evaluations of the new mixed-integer SQP code is between 240 and 500 including those needed for one- or two-sided approximations of partial derivatives or descent directions, respectively. In addition, we present numerical results for a set of 55 test problems with some practical background in petroleum engineering.     

Global optimization of mixed-integer bilevel programming problems

Two approaches that solve the mixed-integer nonlinear bilevel programming problem to global optimality are introduced. The first addresses problems mixed-integer nonlinear in outer variables and C²-nonlinear in inner variables. The second adresses problems with general mixed-integer nonlinear functions in outer level. Inner level functions may be mixed-integer nonlinear in outer variables, linear, polynomial, or multilinear in inner integer variables, and linear in inner continuous variables. This second approach is based on reformulating the mixed-integer inner problem as continuous via its vertex polyheral convex hull representation and solving the resulting nonlinear bilevel optimization problem by a novel deterministic global optimization framework. Computational studies illustrate proposed approaches.     

Reducing the number of variables in integer quadratic programming problem

In this paper, a new variable reduction technique is presented for general integer quadratic programming problem (GP), under which some variables of (GP) can be fixed at zero without sacrificing optimality. A sufficient condition and a necessary condition for the identification of dominated terms are provided. By comparing the given data of the problem and the upper bound of the variables, if they meet certain conditions, some variables can be fixed at zero. We report a computational study to demonstrate the efficacy of the proposed technique in solving general integer quadratic programming problems. Furthermore, we discuss separable integer quadratic programming problems in a simpler and clearer form.     

Conic mixed-integer rounding cuts

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either second-order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree. Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming. We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean–variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability. This research has been supported, in part, by Grant # DMI0700203 from the National Science Foundation.     

Mixed-integer quadratic programming

This paper considers mixed-integer quadratic programs in which the objective function is quadratic in the integer and in the continuous variables, and the constraints are linear in the variables of both types. The generalized Benders' decomposition is a suitable approach for solving such programs. However, the program does not become more tractable if this method is used, since Benders' cuts are quadratic in the integer variables. A new equivalent formulation that renders the program tractable is developed, under which the dual objective function is linear in the integer variables and the dual constraint set is independent of these variables. Benders' cuts that are derived from the new formulation are linear in the integer variables, and the original problem is decomposed into a series of integer linear master problems and standard quadratic subproblems. The new formulation does not introduce new primary variables or new constraints into the computational steps of the decomposition algorithm.The author wishes to thank two anonymous referees for their helpful comments and suggestions for revising the paper.     

Progressive hedging and tabu search applied to mixed integer (0,1) multistage stochastic programming

Many problems faced by decision makers are characterized by a multistage decision process with uncertainty about the future and some decisions constrained to take on values of either zero or one (for example, either open a facility at a location or do not open it). Although some mathematical theory exists concerning such problems, no general-purpose algorithms have been available to address them. In this article, we introduce the first implementation of general purpose methods for finding good solutions to multistage, stochastic mixed-integer (0, 1) programming problems. The solution method makes use of Rockafellar and Wets' progressive hedging algorithm that averages solutions rather than data. Solutions to the induced quadratic (0,1) mixed-integer subproblems are obtained using a tabu search algorithm. We introduce the notion of integer convergence for progressive hedging. Computational experiments verify that the method is effective. The software that we have developed reads standard (SMPS) data files.     

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.     

A new variable reduction technique for convex integer quadratic programs

Convex integer quadratic programming involves minimization of a convex quadratic objective function with affine constraints and is a well-known NP-hard problem with a wide range of applications. We proposed a new variable reduction technique for convex integer quadratic programs (IQP). Based on the optimal values to the continuous relaxation of IQP and a feasible solution to IQP, the proposed technique can be applied to fix some decision variables of an IQP simultaneously at zero without sacrificing optimality. Using this technique, computational effort needed to solve IQP can be greatly reduced. Since a general convex bounded IQP (BIQP) can be transformed to a convex IQP, the proposed technique is also applicable for the convex BIQP. We report a computational study to demonstrate the efficacy of the proposed technique in solving quadratic knapsack problems.     

A Local Relaxation Approach for the Siting of Electrical Substations

The siting and sizing of electrical substations on a rectangular electrical grid can be formulated as an integer programming problem with a quadratic objective and linear constraints. We propose a novel approach that is based on solving a sequence of local relaxations of the problem for a given number of substations. Two methods are discussed for determining a new location from the solution of the relaxed problem. Each leads to a sequence of strictly improving feasible integer solutions. The number of substations is then modified to seek a further reduction in cost. Lower bounds for the solution are also provided by solving a sequence of mixed-integer linear programs. Results are provided for a variety of uniform and Gaussian load distributions as well as some real examples from an electric utility. The results of gams/dicopt, gams/sbb, gams/baron and cplex applied to these problems are also reported. Our algorithm shows slow growth in computational effort with the number of integer variables.     

On approximation algorithms for concave mixed-integer quadratic programming

Concave mixed-integer quadratic programming is the problem of minimizing a concave quadratic polynomial over the mixed-integer points in a polyhedral region. In this work we describe an algorithm that finds an \(\epsilon \)-approximate solution to a concave mixed-integer quadratic programming problem. The running time of the proposed algorithm is polynomial in the size of the problem and in \(1/\epsilon \), provided that the number of integer variables and the number of negative eigenvalues of the objective function are fixed. The running time of the proposed algorithm is expected unless \(\mathcal {P}=\mathcal {NP}\).     

对带有盒约束的二次整数规划的一种线性化方法

本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合0-1整数规划问题,然后利用Ilog-cplex或Excel软件中的规划求解工具进行求解,从而解决原二次整数规划.     

Global optimization of signomial mixed-integer nonlinear programming problems with free variables

Mixed-integer nonlinear programming (MINLP) problems involving general constraints and objective functions with continuous and integer variables occur frequently in engineering design, chemical process industry and management. Although many optimization approaches have been developed for MINLP problems, these methods can only handle signomial terms with positive variables or find a local solution. Therefore, this study proposes a novel method for solving a signomial MINLP problem with free variables to obtain a global optimal solution. The signomial MINLP problem is first transformed into another one containing only positive variables. Then the transformed problem is reformulated as a convex mixed-integer program by the convexification strategies and piecewise linearization techniques. A global optimum of the signomial MINLP problem can finally be found within the tolerable error. Numerical examples are also presented to demonstrate the effectiveness of the proposed method.     

Computing exact solution to nonlinear integer programming: Convergent Lagrangian and objective level cut method

In this paper, we propose a convergent Lagrangian and objective level cut method for computing exact solution to two classes of nonlinear integer programming problems: separable nonlinear integer programming and polynomial zero-one programming. The method exposes an optimal solution to the convex hull of a revised perturbation function by successively reshaping or re-confining the perturbation function. The objective level cut is used to eliminate the duality gap and thus to guarantee the convergence of the Lagrangian method on a revised domain. Computational results are reported for a variety of nonlinear integer programming problems and demonstrate that the proposed method is promising in solving medium-size nonlinear integer programming problems.     

Parametric global optimisation for bilevel programming

We propose a global optimisation approach for the solution of various classes of bilevel programming problems (BLPP) based on recently developed parametric programming algorithms. We first describe how we can recast and solve the inner (follower’s) problem of the bilevel formulation as a multi-parametric programming problem, with parameters being the (unknown) variables of the outer (leader’s) problem. By inserting the obtained rational reaction sets in the upper level problem the overall problem is transformed into a set of independent quadratic, linear or mixed integer linear programming problems, which can be solved to global optimality. In particular, we solve bilevel quadratic and bilevel mixed integer linear problems, with or without right-hand-side uncertainty. A number of examples are presented to illustrate the steps and details of the proposed global optimisation strategy.     

A simple plant-location model for quantity-setting firms subject to price uncertainty

The uncapacitated plant location problem under uncertainty is formulated in a mean-variance framework with prices in various markets correlated via their response to a common random factor. This formulation results in a mixed-integer quadratic programming problem. However, for a given integer solution, the resulting quadratic programming problem is amenable to a very simple solution procedure. The simplicity of this algorithm means that reasonably large problems should be solvable using existing branch-and-bound techniques.     

Comparative studies on dynamic programming and integer programming approaches for concave cost production/inventory control problems

This paper is concerned with classical concave cost multi-echelon production/inventory control problems studied by W. Zangwill and others. It is well known that the problem with m production steps and n time periods can be solved by a dynamic programming algorithm in O(n ⁴ m) steps, which is considered as the fastest algorithm for solving this class of problems. In this paper, we will show that an alternative 0–1 integer programming approach can solve the same problem much faster particularly when n is large and the number of 0–1 integer variables is relatively few. This class of problems include, among others problem with set-up cost function and piecewise linear cost function with fewer linear pieces. The new approach can solve problems with mixed concave/convex cost functions, which cannot be solved by dynamic programming algorithms.     

A distributed computation algorithm for solving portfolio problems with integer variables

A portfolio problem with integer variables can facilitate the use of complex models, including models containing discrete asset values, transaction costs, and logical constraints. This study proposes a distributed algorithm for solving a portfolio program to obtain a global optimum. For a portfolio problem with n integer variables, the objective function first is converted into an ellipse function containing n separated quadratic terms. Next, the problem is decomposed into m equal-size separable programming problems solvable by a distributed computation system composed of m personal computers linked via the Internet. The numerical examples illustrate that the proposed method can obtain the global optimum effectively for large scale portfolio problems involving integral variables.     

Solving mixed-integer robust optimization problems with interval uncertainty using Benders decomposition

Uncertainty and integer variables often exist together in economics and engineering design problems. The goal of robust optimization problems is to find an optimal solution that has acceptable sensitivity with respect to uncertain factors. Including integer variables with or without uncertainty can lead to formulations that are computationally expensive to solve. Previous approaches for robust optimization problems under interval uncertainty involve nested optimization or are not applicable to mixed-integer problems where the objective or constraint functions are neither quadratic, nor linear. The overall objective in this paper is to present an efficient robust optimization method that does not contain nested optimization and is applicable to mixed-integer problems with quasiconvex constraints (? type) and convex objective funtion. The proposed method is applied to a variety of numerical examples to test its applicability and numerical evidence is provided for convergence in general as well as some theoretical results for problems with linear constraints.     

An SQP algorithm for extended linear-quadratic problems in stochastic programming

Extended Linear-Quadratic Programming (ELQP) problems were introduced by Rockafellar and Wets for various models in stochastic programming and multistage optimization. Several numerical methods with linear convergence rates have been developed for solving fully quadratic ELQP problems, where the primal and dual coefficient matrices are positive definite. We present a two-stage sequential quadratic programming (SQP) method for solving ELQP problems arising in stochastic programming. The first stage algorithm realizes global convergence and the second stage algorithm realizes superlinear local convergence under a condition calledB-regularity.B-regularity is milder than the fully quadratic condition; the primal coefficient matrix need not be positive definite. Numerical tests are given to demonstrate the efficiency of the algorithm. Solution properties of the ELQP problem underB-regularity are also discussed.Supported by the Australian Research Council.     

Threshold Boolean form for joint probabilistic constraints with random technology matrix

We develop a new modeling and solution method for stochastic programming problems that include a joint probabilistic constraint in which the multirow random technology matrix is discretely distributed. We binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined Boolean function (pdBf) representing the probabilistic constraint. We then construct a tight threshold Boolean minorant for the pdBf. Any separating structure of the tight threshold Boolean minorant defines sufficient conditions for the satisfaction of the probabilistic constraint and takes the form of a system of linear constraints. We use the separating structure to derive three new deterministic formulations for the studied stochastic problem, and we derive a set of strengthening valid inequalities. A crucial feature of the new integer formulations is that the number of integer variables does not depend on the number of scenarios used to represent uncertainty. The computational study, based on instances of the stochastic capital rationing problem, shows that the mixed-integer linear programming formulations are orders of magnitude faster to solve than the mixed-integer nonlinear programming formulation. The method integrating the valid inequalities in a branch-and-bound algorithm has the best performance.