Boosting the feasibility pump

The feasibility pump (FP) has proved to be an effective method for finding feasible solutions to mixed integer programming problems. FP iterates between a rounding procedure and a projection procedure, which together provide a sequence of points alternating between LP feasible but fractional solutions, and integer but LP infeasible solutions. The process attempts to minimize the distance between consecutive iterates, producing an integer feasible solution when closing the distance between them. We investigate the benefits of enhancing the rounding procedure with a clever integer line search that efficiently explores a large set of integer points. An extensive computational study on benchmark instances demonstrates the efficacy of the proposed approach.     

Converging upon basic feasible solutions through Dantzig–Wolfe decomposition

We derive an important property for solving large-scale integer programs by examining the master problem in Dantzig–Wolfe decomposition. In particular, we prove that if a linear program can be divided into subproblems with affinely independent corner points, then there is a direct mapping between basic feasible solutions in the master and original problems. This has implications for integer programs where the feasible region has integer corner points, ensuring that integer solutions to the original problem will be found even through the decomposition approach. An application to air traffic flow scheduling, which motivated this result, is highlighted.     

Ranking in quadratic integer programming problems

The present paper develops an algorithm for ranking the integer feasible solutions of a quadratic integer programming (QIP) problem. A linear integer programming (LIP) problem is constructed which provides bounds on the values of the objective function of the quadratic problem. The integer feasible solutions of this related integer linear programming problem are systematically scanned to rank the integer feasible solutions of the quadratic problem in non-decreasing order of the objective function values. The ranking in the QIP problem is useful in solving a nonlinear integer programming problem in which some other complicated nonlinear restrictions are imposed which cannot be included in the simple linear constraints of QIP, the objective function being still quadratic.     

Nondominated Nash points: application of biobjective mixed integer programming

We study the connection between biobjective mixed integer linear programming and normal form games with two players. We first investigate computing Nash equilibria of normal form games with two players using single-objective mixed integer linear programming. Then, we define the concept of efficient (Pareto optimal) Nash equilibria. This concept is precisely equivalent to the concept of efficient solutions in multi-objective optimization, where the solutions are Nash equilibria. We prove that the set of all points in the payoff (or objective) space of a normal form game with two players corresponding to the utilities of players in an efficient Nash equilibrium, the so-called nondominated Nash points, is finite. We demonstrate that biobjective mixed integer linear programming, where the utility of each player is an objective function, can be used to compute the set of nondominated Nash points. Finally, we illustrate how the nondominated Nash points can be used to determine the disagreement point of a bargaining problem.     

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.     

Parametric Integer Programming Analysis: A Contraction Approach

An algorithm is presented for solving families of integer linear programming problems in which the problems are "related" by having identical objective coefficients and constraint matrix coefficients. The righthand-side constants have the form b + θd where b and d are conformable vectors and θ varies from zero to one.The approach consists primarily of solving the most relaxed problem (θ = 1) using cutting planes and then contracting the region of feasible integer solutions in such a manner that the current optimal integer solution is eliminated.The algorithm was applied to 1800 integer linear programming problems with reasonable success. Integer programming problems which have proved to be unsolvable using cutting planes have been solved by expanding the region of feasible integer solutions (θ = 1) and then contracting to the original region.     

A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints

In this paper, we consider a mathematical program with complementarity constraints. We present a modified relaxed program for this problem, which involves less constraints than the relaxation scheme studied by Scholtes (2000). We show that the linear independence constraint qualification holds for the new relaxed problem under some mild conditions. We also consider a limiting behavior of the relaxed problem. We prove that any accumulation point of stationary points of the relaxed problems is C-stationary to the original problem under the MPEC linear independence constraint qualification and, if the Hessian matrices of the Lagrangian functions of the relaxed problems are uniformly bounded below on the corresponding tangent space, it is M-stationary. We also obtain some sufficient conditions of B-stationarity for a feasible point of the original problem. In particular, some conditions described by the eigenvalues of the Hessian matrices mentioned above are new and can be verified easily. This work was supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science, Sports, and Culture of Japan. The authors are grateful to an anonymous referee for critical comments.     

SO-I: a surrogate model algorithm for expensive nonlinear integer programming problems including global optimization applications

This paper presents the surrogate model based algorithm SO-I for solving purely integer optimization problems that have computationally expensive black-box objective functions and that may have computationally expensive constraints. The algorithm was developed for solving global optimization problems, meaning that the relaxed optimization problems have many local optima. However, the method is also shown to perform well on many local optimization problems, and problems with linear objective functions. The performance of SO-I, a genetic algorithm, Nonsmooth Optimization by Mesh Adaptive Direct Search (NOMAD), SO-MI (Müller et al. in Comput Oper Res 40(5):1383–1400, 2013), variable neighborhood search, and a version of SO-I that only uses a local search has been compared on 17 test problems from the literature, and on eight realizations of two application problems. One application problem relates to hydropower generation, and the other one to throughput maximization. The numerical results show that SO-I finds good solutions most efficiently. Moreover, as opposed to SO-MI, SO-I is able to find feasible points by employing a first optimization phase that aims at minimizing a constraint violation function. A feasible user-supplied point is not necessary.     

组稀疏优化问题精确连续Capped-L_(1)松弛北大核心CSCD

本文主要研究损失函数为凸函数且带有约束的组稀疏正则回归问题及组稀疏正则项的精确连续Capped-L_(1)松弛问题.首先对组Capped-L_(1)松弛问题定义了三类稳定点:D(irectional)-稳定点、C(ritical)-稳定点、L(ifted)-稳定点,然后刻画了这三类稳定点之间的关系.进一步,给出了组Capped-L_(1)松弛问题和原始组稀疏正则问题的最优性条件,并从全局解和局部解角度讨论了松弛问题和原问题解的等价关系.     

Single machine scheduling with symmetric earliness and tardiness penalties

This research focuses on scheduling jobs with varying processing times and distinct due dates on a single machine subject to earliness and tardiness penalties. Hence, this work will find application in a just-in-time (JIT) production environment. The scheduling problem is formulated as a 0–1 linear integer program with three sets of constraints, where the objective is to minimize the sum of the absolute deviations between job completion times and their respective due dates. The first two sets of constraints are equivalent to the supply and demand constraints of an assignment problem. The third set, which represents the process time non-overlap constraints, is relaxed to form the Lagrangian dual problem. The dual problem is then solved using the subgradient algorithm. Efficient heuristics have also been developed in this work to yield initial primal feasible solutions and to convert primal infeasible solutions to feasibility. The computational results show that the relative deviation from optimality obtained by the subgradient algorithm is less than 3% for problem sizes varying from 10 to 100 jobs.     

Discrete location for bundled demand points

This paper considers a discrete location problem where the demand points are grouped. We propose a formulation, an enforcement for it, and an associated Lagrangian relaxation, and then we build feasible solutions to the problem from the optimal solutions to the relaxed subproblems. Valid inequalities for the formulation are also identified and added to the set of relaxed constraints. This method produces good feasible solutions and enables us to address large instances of the problem. Computational experiments have been performed with benchmark instances from the literature on related problems.     

Feasibility pump 2.0

Finding a feasible solution of a given mixed-integer programming (MIP) model is a very important ${\mathcal{NP}}$ -complete problem that can be extremely hard in practice. Feasibility Pump (FP) is a heuristic scheme for finding a feasible solution to general MIPs that can be viewed as a clever way to round a sequence of fractional solutions of the LP relaxation, until a feasible one is eventually found. In this paper we study the effect of replacing the original rounding function (which is fast and simple, but somehow blind) with more clever rounding heuristics. In particular, we investigate the use of a diving-like procedure based on rounding and constraint propagation—a basic tool in Constraint Programming. Extensive computational results on binary and general integer MIPs from the literature show that the new approach produces a substantial improvement of the FP success rate, without slowing-down the method and with a significantly better quality of the feasible solutions found.     

New Relaxation Method for Mathematical Programs with Complementarity Constraints

In this paper, we present a new relaxation method for mathematical programs with complementarity constraints. Based on the fact that a variational inequality problem defined on a simplex can be represented by a finite number of inequalities, we use an expansive simplex instead of the nonnegative orthant involved in the complementarity constraints. We then remove some inequalities and obtain a standard nonlinear program. We show that the linear independence constraint qualification or the Mangasarian–Fromovitz constraint qualification holds for the relaxed problem under some mild conditions. We consider also a limiting behavior of the relaxed problem. We prove that any accumulation point of stationary points of the relaxed problems is a weakly stationary point of the original problem and that, if the function involved in the complementarity constraints does not vanish at this point, it is C-stationary. We obtain also some sufficient conditions of B-stationarity for a feasible point of the original problem. In particular, some conditions described by the eigenvalues of the Hessian matrices of the Lagrangian functions of the relaxed problems are new and can be verified easily. Our limited numerical experience indicates that the proposed approach is promising.     

Heuristics Based on Partial Enumeration for the Unrelated Parallel Processor Scheduling Problem

The classical deterministic scheduling problem of minimizing the makespan on unrelated parallel processors is known to be NP-hard in the strong sense. Given the mixed integer linear model with binary decision variables, this paper presents heuristic algorithms based on partial enumeration. Basically, they consist in the construction of mixed integer subproblems, considering the integrality of some subset of variables, formulated using the information obtained from the solution of the linear relaxed problem. Computational experiments are reported for a collection of test problems, showing that some of the proposed algorithms achieve better solutions than other relevant approximation algorithms published up to now.     

求解互补约束优化问题的乘子松弛法

利用互补问题的Lagrange函数, 给出了互补约束优化问题\,(MPCC)\,的一种新松弛问题. 在较弱的条件下, 新松弛问题满足线性独立约束规范. 在此基础上, 提出了求解互补约束优化问题的乘子松弛法. 在MPCC-LICQ条件下, 松弛问题稳定点的任何聚点都是MPCC的M-稳定点. 无需二阶必要条件, 只在ULSC条件下, 就可保证聚点是MPCC的B-稳定点. 另外, 给出了算法收敛于B-稳定点的新条件.     

GOSAC: global optimization with surrogate approximation of constraints

We introduce GOSAC, a global optimization algorithm for problems with computationally expensive black-box constraints and computationally cheap objective functions. The variables may be continuous, integer, or mixed-integer. GOSAC uses a two-phase optimization approach. The first phase aims at finding a feasible point by solving a multi-objective optimization problem in which the constraints are minimized simultaneously. The second phase aims at improving the feasible solution. In both phases, we use cubic radial basis function surrogate models to approximate the computationally expensive constraints. We iteratively select sample points by minimizing the computationally cheap objective function subject to the constraint function approximations. We assess GOSAC’s efficiency on computationally cheap test problems with integer, mixed-integer, and continuous variables and two environmental applications. We compare GOSAC to NOMAD and a genetic algorithm (GA). The results of the numerical experiments show that for a given budget of allowed expensive constraint evaluations, GOSAC finds better feasible solutions more efficiently than NOMAD and GA for most benchmark problems and both applications. GOSAC finds feasible solutions with a higher probability than NOMAD and GOSAC.     

Generalized dynamic programming methods in integer programming

When regarded as a shortest route problem, an integer program can be seen to have a particularly simple structure. This allows the development of an algorithm for finding thek ^th best solution to an integer programming problem with max{O(kmn), O(k logk)} operations. Apart from its value in the parametric study of an optimal solution, the approach leads to a general integer programming algorithm consisting of (1) problem relaxation, (2) solution of the relaxed problem parametrically by dynamic programming, and (3) generation ofk ^th best solutions until a feasible solution is found. Elementary methods based on duality for reducingk for a given problem relaxation are then outlined, and some examples and computational aspects are discussed.     

An interior point algorithm to solve computationally difficult set covering problems

We present an interior point approach to the zero–one integer programming feasibility problem based on the minimization of a nonconvex potential function. Given a polytope defined by a set of linear inequalities, this procedure generates a sequence of strict interior points of this polytope, such that each consecutive point reduces the value of the potential function. An integer solution (not necessarily feasible) is generated at each iteration by a rounding scheme. The direction used to determine the new iterate is computed by solving a nonconvex quadratic program on an ellipsoid. We illustrate the approach by considering a class of difficult set covering problems that arise from computing the 1-width of the incidence matrix of Steiner triple systems.     

Upper bounds on the average number of iterations for some algorithms of solving the set packing problem

The set packing problem and the corresponding integer linear programming model are considered. Using the regular partitioning method and available estimates of the average number of feasible solutions of this problem, upper bounds on the average number of iterations for the first Gomory method, the branch-and-bound method (the Land and Doig scheme), and the L-class enumeration algorithm are obtained. The possibilities of using the proposed approach for other integer programs are discussed.     

Strictly feasible solutions and strict complementarity in multiple objective linear optimization

Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established. Here, we use these results to prove some relationships between multiple objective linear primal and dual problems. We extend the available results on single objective linear primal and dual problems to multiple objective linear primal and dual problems. Complementary slackness conditions for efficient solutions, and conditions for the existence of weakly efficient solution sets and existence of strictly primal and dual feasible points are established. We show that primal-dual (weakly) efficient solutions satisfying strictly complementary conditions exist. Furthermore, we consider Isermann’s and Kolumban’s dual problems and establish conditions for the existence of strictly primal and dual feasible points. We show the existence of primal-dual feasible points satisfying strictly complementary conditions for Isermann’s dual problem. Also, we give an alternative proof to establish necessary conditions for weakly efficient solutions of multiple objective programs, assuming the Kuhn–Tucker (KT) constraint qualification. We also provide a new condition to ensure the KT constraint qualification.