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.     

Some relationships between lagrangian and surrogate duality in integer programming

Lagrangian dual approaches have been employed successfully in a number of integer programming situations to provide bounds for branch-and-bound procedures. This paper investigates some relationship between bounds obtained from lagrangian duals and those derived from the lesser known, but theoretically more powerful surrogate duals. A generalization of Geoffrion's integrality property, some complementary slackness relationships between optimal solutions, and some empirical results are presented and used to argue for the relative value of surrogate duals in integer programming. These and other results are then shown to lead naturally to a two-phase algorithm which optimizes first the computationally easier lagrangian dual and then the surrogate dual.     

Alternating direction augmented Lagrangian methods for semidefinite programming

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our algorithms are robust and very efficient due to their ability or exploit special structures, such as sparsity and constraint orthogonality in these problems.     

Lagrangean decomposition for integer nonlinear programming with linear constraints

We present a Lagrangean decomposition to study integer nonlinear programming problems. Solving the dual Lagrangean relaxation we have to obtain at each iteration the solution of a nonlinear programming with continuous variables and an integer linear programming. Decreasing iteratively the primal—dual gap we propose two algorithms to treat the integer nonlinear programming.This work was partially supported by CNPq and FINEP.     

Extension of M-Convexity and L-Convexity to Polyhedral Convex Functions

The concepts of M-convex and L-convex functions were proposed by Murota in 1996 as two mutually conjugate classes of discrete functions over integer lattice points. M/L-convex functions are deeply connected with the well-solvability in nonlinear combinatorial optimization with integer variables. In this paper, we extend the concept of M-convexity and L-convexity to polyhedral convex functions, aiming at clarifying the well-behaved structure in well-solved nonlinear combinatorial optimization problems in real variables. The extended M/L-convexity often appears in nonlinear combinatorial optimization problems with piecewise-linear convex cost. We investigate the structure of polyhedral M-convex and L-convex functions from the dual viewpoint of analysis and combinatorics and provide some properties and characterizations. It is also shown that polyhedral M/L-convex functions have nice conjugacy relationships.     

Dual decomposition in stochastic integer programming

We present an algorithm for solving stochastic integer programming problems with recourse, based on a dual decomposition scheme and Lagrangian relaxation. The approach can be applied to multi-stage problems with mixed-integer variables in each time stage. Numerical experience is presented for some two-stage test problems.     

An elementary survey of general duality theory in mathematical programming

We survey some recent developments in duality theory with the idea of explaining and unifying certain basic duality results in both nonlinear and integer programming. The idea of replacing dual variables (prices) by price functions, suggested by Everett and developed by Gould, is coupled with an appropriate dual problem with the consequence that many of the results resemble those used in linear programming. The dual problem adopted has a (traditional) economic interpretation and dual feasibility then provides a simple alternative to concepts such as conjugate functions or subdifferentials used in the study of optimality. In addition we attempt to make precise the relationship between primal, dual and saddlepoint results in both the traditional Lagrangean and the more general duality theories and to see the implications of passing from prices to price functions. Finally, and perhaps surprisingly, it appears that all the standard algorithms terminate by constructing primal and dual feasible solutions of equal value, i.e., by satisfying generalised optimality conditions.     

CALCULATION OF PENALTIES IN ALGORITHM OF MIXED INTEGER PROGRAMMING SOLVING WITH REVISED DUAL SIMPLEX METHOD FOR BOUNDED VARIABLES

1.IntroductionThestudiesonthebranch--and--boundalgorithmofintegerprogramminghavebeencarriedoutsince60's.TheeffortsinimprovingthealgorithmaremainlyconcentratedonspeedinguptherelatedLPsolutionforeachnodeandmakingbetterselectionofnodeandbranchforexaming...     

Duality for Algebraic Linear Programming

A duality theory for algebraic linear (integer) programming (ALP) is developed which is of the same importance for linear (integer) programming with linear algebraic objectives as linear programming duality is for classical LP. In particular, optimality criteria for primal, primal-dual, and dual methods are given which generalize feasibility and complementarity criteria of classical LP. Strong duality results are given for special combinatorial problems. Further, the validity and finiteness of a primal simplex method based on a feasibility criterion are proved in the case of nondiscrete variables. In this case a strong duality result is shown.     

The discrete moment problem with fractional moments

Discrete moment problems (DMP) with integer moments were first introduced by Prékopa to provide sharp lower and upper bounds for functions of discrete random variables. Prékopa also developed fast and stable dual type linear programming methods for the numerical solutions of the problem. In this paper, we assume that some fractional moments are also available and propose basic theory and a solution method for the bounding problems. Numerical experiments show significant improvement in the tightness of the bounds.     

Integer programming and pricing revisited

Three applications of duality are mentioned: mathematical, computational,and economic. One of the earliest attempts toproduce a dualof an integer programme with economic interpretations was byGomory & Baumol in 1960. This is describedtogether withits economic properties and some refinements and corrections.A more recent integer programming dual due to Chvátal,whose main use to date has been computational, is then described.It is shown that this can be given an economic interpretationas a generalization of Gomory & Baumol‘s dual whichrectifies some of the deficiencies of the latter. The computationalproblems of calculating Chvátal’s dual are remarkedon.     

整数规划新进展

整数规划是对全部或部分决策变量为整数的最优化问题的模型、算法及应用等的研究, 是运筹学和管理科学中应用最广泛的优化模型之一. 首先简要回顾整数规划的历史和发展进程, 概述线性和非线性整数规划的一些经典方法. 然后着重讨论整数规划若干新进展, 包括0-1二次规划的半定规划~(SDP)~松弛和随机化方法, 带半连续变量和稀疏约束的优化问题的整数规划模型和方法, 以及0-1二次规划的协正锥规划表示和协正锥的层级半定规划~(SDP)~逼近. 最后, 对整数规划未来研究方向进行展望并对一些公开问题进行讨论.     

Deriving compact extended formulations via LP-based separation techniques

The best formulations for some combinatorial optimization problems are integer linear programming models with an exponential number of rows and/or columns, which are solved incrementally by generating missing rows and columns only when needed. As an alternative to row generation, some exponential formulations can be rewritten in a compact extended form, which have only a polynomial number of constraints and a polynomial, although larger, number of variables. As an alternative to column generation, there are compact extended formulations for the dual problems, which lead to compact equivalent primal formulations, again with only a polynomial number of constraints and variables. In this this paper we introduce a tool to derive compact extended formulations and survey many combinatorial optimization problems for which it can be applied. The tool is based on the possibility of formulating the separation procedure by an LP model. It can be seen as one further method to generate compact extended formulations besides other tools of geometric and combinatorial nature present in the literature.     

Determination of optimal vertices from feasible solutions in unimodular linear programming

In this paper we consider a linear programming problem with the underlying matrix unimodular, and the other data integer. Given arbitrary near optimum feasible solutions to the primal and the dual problems, we obtain conditions under which statements can be made about the value of certain variables in optimal vertices. Such results have applications to the problem of determining the stopping criterion in interior point methods like the primal—dual affine scaling method and the path following methods for linear programming.This author's research is partially supported by NSF grant DDM-8921835 and Airforce Grant AFSOR-88-0088.     

On discrete lot-sizing and scheduling on identical parallel machines

We consider the multi-item discrete lot-sizing and scheduling problem on identical parallel machines. Based on the fact that the machines are identical, we introduce aggregate integer variables instead of individual variables for each machine. For the problem with start-up costs, we show that the inequalities based on a unit flow formulation for each machine can be replaced by a single integer flow formulation without any change in the resulting LP bound. For the resulting integer lot-sizing with start-ups subproblem, we show how inequalities for the unit demand case can be generalized and how an approximate version of the extended formulation of Eppen and Martin can be constructed. The results of some computational experiments carried out to compare the effectiveness of the various mixed-integer programming formulations are presented.     

Strengthening lattice-free cuts using non-negativity

In recent years there has been growing interest in generating valid inequalities for mixed-integer programs using sets with two or more constraints. In particular, Andersen et al. (2007) [2] and Borozan and Cornuéjols (2009) [3] have studied sets defined by equations that contain exactly one integer variable per row. The integer variables are not restricted in sign. Cutting planes based on this approach have already been computationally studied by Espinoza (2008) [8] for general mixed-integer problems, and there is ongoing computational research in this area.In this paper, we extend the model studied in the earlier papers and require the integer variables to be non-negative. We extend the results in [2] and [3] to our case, and show that cuts generated by their approach can be strengthened by using the non-negativity of the integer variables. In particular, it is possible to obtain cuts which have negative coefficients for some variables.     

Energy contracts management by stochastic programming techniques

We consider the problem of optimal management of energy contracts, with bounds on the local (time step) amounts and global (whole period) amounts to be traded, integer constraint on the decision variables and uncertainty on prices only. After building a finite state Markov chain by using vectorial quantization tree method, we rely on the stochastic dual dynamic programming (SDDP) method to solve the continuous relaxation of this stochastic optimization problem. An heuristic for computing sub optimal solutions to the integer optimization problem, based on the Bellman values of the continuous relaxation, is provided. Combining the previous techniques, we are able to deal with high-dimensional state variables problems. Numerical tests applied to realistic energy markets problems have been performed.     

Symmetric duality for minimax mixed integer programming problems with pseudo-invexity

In this paper, we consider the pair of symmetric dual multiobjective variational mixed integer programs proposed by Chen and Yang [X. Chen, J. Yang, Symmetric duality for minimax multiobjective variational mixed integer programming problems with partial-invexity, European Journal of Operational Research 181 (2007) 76-85.] and extend some of their results under the assumptions of partial-pseudo-invexity and separability on the functions involved. These results include several results available in literature as special cases.     

Efficient market-clearing prices in markets with nonconvexities

This paper addresses the existence of market clearing prices and the economic interpretation of strong duality for integer programs in the economic analysis of markets with nonconvexities (indivisibilities). Electric power markets in which nonconvexities arise from the operating characteristics of generators motivate our analysis; however, the results presented here are general and can be applied to other markets in which nonconvexities are important. We show that the optimal solution to a linear program that solves the mixed integer program has dual variables that: (1) have the traditional economic interpretation as prices; (2) explicitly price integral activities; and (3) clear the market in the presence of nonconvexities. We then show how this methodology can be used to interpret the solutions to nonconvex problems such as the problem discussed by Scarf [Journal of Economic Perspectives 8(4) (1994) 111].     

Operations that preserve total dual integrality

There are many useful operations, such as adding slack variables, taking scalar multiples of inequalities, and applying Fourier-Motzkin elimination, that can be performed on a linear system such that if the system defines an integer polyhedron then so does the derived system. The topic dealt with here is whether or not these operations also preserve total dual integrality of linear systems.