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.     

Classification of Dantzig–Wolfe reformulations for binary mixed integer programming problems

In this note, we provide a classification of Dantzig–Wolfe reformulations for Binary Mixed Integer Programming Problems. We specifically focus on modeling the binary conditions in the convexification approach to the Dantzig–Wolfe decomposition. For a general Binary Mixed Integer Programming problem, an extreme point of the overall problem does not necessarily correspond to an extreme point of the subproblem. Therefore, the binary conditions cannot in general be imposed on the new master problem variables but must be imposed on the original binary variables. In some cases, however, it is possible to impose the binary restrictions directly on the new master problem variables. The issue of imposing binary conditions on the original variables versus the master problem variables has not been discussed systematically for MIP problems in general in the literature and most of the research has been focused on the pure binary case. The classification indicates in which cases you can, and cannot, impose binary conditions on the new master problem variables.     

Chvátal closures for mixed integer programming problems

Chvátal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cutting-plane proofs is that for a polyhedronP and integral vectorw, if max(wx|x P, wx integer} =t, thenwx t is valid for all integral vectors inP. We consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that be integer for some other integral vector . The cutting-plane proofs thus obtained may be seen either as an abstraction of Gomory's mixed integer cutting-plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedronP, the set of vectors that satisfy every cutting plane forP with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chvátal closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.Supported by Sonderforschungsbereich 303 (DFG) and by NSF Grant Number ECS-8611841.Supported by NSF Grant Number ECS-8418392 and Sonderforschungsbereich 303 (DFG), Institut für Ökonometrie und Operations Research, Universität Bonn, FR Germany.     

On the exactness of the ε-constraint method for biobjective nonlinear integer programming

The ε-constraint method is a well-known scalarization technique used for multiobjective optimization. We explore how to properly define the step size parameter of the method in order to guarantee its exactness when dealing with biobjective nonlinear integer problems. Under specific assumptions, we prove that the number of subproblems that the method needs to address to detect the complete Pareto front is finite. We report numerical results on portfolio optimization instances built on real-world data and show a comparison with an existing criterion space algorithm.     

A Lagrange–Newton algorithm for sparse nonlinear programming

Mathematical Programming - The sparse nonlinear programming (SNP) problem has wide applications in signal and image processing, machine learning and finance, etc. However, the computational...     

A composite collocation method for the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations

This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations. The method is based on the composite collocation method. The properties of hybrid of block-pulse functions and Lagrange polynomials are discussed and utilized to define the composite interpolation operator. The estimates for the errors are given. The composite interpolation operator together with the Gaussian integration formula are then used to transform the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations into a system of nonlinear equations. The efficiency and accuracy of the proposed method is illustrated by four numerical examples.     

A primal–dual interior point method for nonlinear semidefinite programming

This paper is concerned with a primal–dual interior point method for solving nonlinear semidefinite programming problems. The method consists of the outer iteration (SDPIP) that finds a KKT point and the inner iteration (SDPLS) that calculates an approximate barrier KKT point. Algorithm SDPLS uses a commutative class of Newton-like directions for the generation of line search directions. By combining the primal barrier penalty function and the primal–dual barrier function, a new primal–dual merit function is proposed. We prove the global convergence property of our method. Finally some numerical experiments are given.     

An optimal repair–replacement policy for a cold standby system with use priority

In this paper, the maintenance problem for a cold standby system consisting of two dissimilar components and one repairman is studied. Assume that both component 1 and component 2 after repair follow geometric process repair and component 1 is given priority in use when both components are workable. Under these assumptions, using geometric process repair model, we consider a replacement policy N under which the system is replaced when the number of failures of component 1 reaches N. Our purpose is to determine an optimal replacement policy N1 such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression for the average cost rate of the system is derived and the corresponding optimal replacement policy N1 can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results and the model applicability.     

A weighted max–min model for fuzzy goal programming

After Narasimhan's pioneering study of applying fuzzy set theory to goal programming in 1980, many achievements in the field have been recorded. Most of them followed the max–min approach. However, when objectives have different levels of importance, only the weighted additive model of Tiwari et al. seems to be applicable. However, the shortcoming of the additive model is that the summation of quasiconcave functions may not be quasiconcave. This study proposes a novel weighted max–min model for fuzzy goal programming (FGP) and for fuzzy multiple objective decision-making. The proposed model adapts well to even the most complicated membership functions. Numerical examples demonstrate that the proposed model can be effectively incorporated with other approaches to FGP and is superior to the weighted additive approach.     

A provably convergent heuristic for stochastic bicriteria integer programming

We propose a general-purpose algorithm APS (Adaptive Pareto-Sampling) for determining the set of Pareto-optimal solutions of bicriteria combinatorial optimization (CO) problems under uncertainty, where the objective functions are expectations of random variables depending on a decision from a finite feasible set. APS is iterative and population-based and combines random sampling with the solution of corresponding deterministic bicriteria CO problem instances. Special attention is given to the case where the corresponding deterministic bicriteria CO problem can be formulated as a bicriteria integer linear program (ILP). In this case, well-known solution techniques such as the algorithm by Chalmet et al. can be applied for solving the deterministic subproblem. If the execution of APS is terminated after a given number of iterations, only an approximate solution is obtained in general, such that APS must be considered a metaheuristic. Nevertheless, a strict mathematical result is shown that ensures, under rather mild conditions, convergence of the current solution set to the set of Pareto-optimal solutions. A modification replacing or supporting the bicriteria ILP solver by some metaheuristic for multicriteria CO problems is discussed. As an illustration, we outline the application of the method to stochastic bicriteria knapsack problems by specializing the general framework to this particular case and by providing computational examples.     

A two-grid method with backtracking for the mixed Navier–Stokes/Darcy model

In this paper, we consider the effect of adding a coarse mesh correction to the two-grid algorithm for the mixed Navier–Stokes/Darcy model. The method yields both L² and H¹ optimal velocity and piezometric head approximations and an L² optimal pressure approximation. The method involves solving one small, coupled, nonlinear coarse mesh problem, two independent subproblems (linear Navier–Stokes equation and Darcy equation) on the fine mesh, and a correction problem on the coarse mesh. Theoretical analysis and numerical tests are done to indicate the significance of this method.     

On a mixture of the fix-and-relax coordination and Lagrangian substitution schemes for multistage stochastic mixed integer programming

We present a framework for solving large-scale multistage mixed 0–1 optimization problems under uncertainty in the coefficients of the objective function, the right-hand side vector, and the constraint matrix. A scenario tree-based scheme is used to represent the Deterministic Equivalent Model of the stochastic mixed 0–1 program with complete recourse. The constraints are modeled by a splitting variable representation via scenarios. So, a mixed 0–1 model for each scenario cluster is considered, plus the nonanticipativity constraints that equate the 0–1 and continuous so-called common variables from the same group of scenarios in each stage. Given the high dimensions of the stochastic instances in the real world, it is not realistic to obtain the optimal solution for the problem. Instead we use the so-called Fix-and-Relax Coordination (FRC) algorithm to exploit the characteristics of the nonanticipativity constraints of the stochastic model. A mixture of the FRC approach and the Lagrangian Substitution and Decomposition schemes is proposed for satisfying, both, the integrality constraints for the 0–1 variables and the nonanticipativity constraints. This invited paper is discussed in the comments available at: doi:, doi:, doi:, doi:.     

bc–opt: a branch-and-cut code for mixed integer programs

A branch-and-cut mixed integer programming system, called bc–opt, is described, incorporating most of the valid inequalities that have been used or suggested for such systems, namely lifted 0-1 knapsack inequalities, 0-1 gub knapsack and integer knapsack inequalities, flowcover and continuous knapsack inequalities, path inequalities for fixed charge network flow structure and Gomory mixed integer cuts. The principal development is a set of interface routines allowing these cut routines to generate cuts for new subsets or aggregations of constraints. The system is built using the XPRESS Optimisation Subroutine Library (XOSL) which includes a cut manager that handles the tree and cut management, so that the user only essentially needs to develop the cut separation routines. Results for the MIPLIB3.0 library are presented - 37 of the instances are solved very easily, optimal or near optimal solution are produced for 18 other instances, and of the 4 remaining instances, 3 have 0, +1, -1 matrices for which bc–opt contains no special features. Received May 11, 1997 / Revised version received March 8, 1999?Published online June 11, 1999     

A cutting-plane approach to mixed 0–1 stochastic integer programs

We consider a mixed 0–1 integer programming problem with dual block-angular structure arising in two-stage stochastic programming. A relaxation is proposed such that the problem is decomposed into subproblems each corresponding to the outcomes of the random variable. The convex hull of feasible solutions for the relaxation is characterized using results from disjunctive programming and it is shown how Lift-and-Project cuts can be generated for one subproblem and made valid for different outcomes.     

Robust regression for mixed Poisson–Gaussian model

This paper focuses on efficient computational approaches to compute approximate solutions of a linear inverse problem that is contaminated with mixed Poisson–Gaussian noise, and when there are additional outliers in the measured data. The Poisson–Gaussian noise leads to a weighted minimization problem, with solution-dependent weights. To address outliers, the standard least squares fit-to-data metric is replaced by the Talwar robust regression function. Convexity, regularization parameter selection schemes, and incorporation of non-negative constraints are investigated. A projected Newton algorithm is used to solve the resulting constrained optimization problem, and a preconditioner is proposed to accelerate conjugate gradient Hessian solves. Numerical experiments on problems from image deblurring illustrate the effectiveness of the methods.     

On the polynomial mixed 0–1 fractional programming problems

This paper proposes a new method of solving polynomial mixed 0–1 fractional programming (P01FP) problems to obtain a global optimum. Given a polynomial 0–1 term x₁x₂,…,x_ny, where x_i is a 0–1 variable and y is a continuous variable; we develop a linearization technique to transfer the x₁x₂,…,x_ny term into a set of mixed 0–1 linear inequalities. Based on this technique, the P01FP can then be solved by a branch-and-bound method to obtain the global solution.     

Asymptotic behavior of integer programming and the stability of the Castelnuovo–Mumford regularity

Mathematical Programming - The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer...     

A Ginzburg–Landau model for the phase transition in Helium II

The paper deals with the second-order phase transition in Helium II by a Ginzburg–Landau model, in which any particle has simultaneously a normal and superfluid velocity. This pattern is able to describe the classical effects of Helium II as the phase diagram, the vortices, the second sound and the thermomechanical effect. Finally, the vorticities and turbulence are described by an extension of the model in which the material time derivative is used.     

A mixed problem for the steady Navier–Stokes equations

We consider a mixed boundary problem for the Navier–Stokes equations in a bounded Lipschitz two-dimensional domain: we assign a Dirichlet condition on the curve portion of the boundary and a slip zero condition on its straight portion. We prove that the problem has a solution provided the boundary datum and the body force belong to a Lebesgue’s space and to the Hardy space respectively.     

An optimal reinsurance problem in the Cramér–Lundberg model

In this article we consider the surplus process of an insurance company within the Cramér–Lundberg framework with the intention of controlling its performance by means of dynamic reinsurance. Our aim is to find a general dynamic reinsurance strategy that maximizes the expected discounted surplus level integrated over time. Using analytical methods we identify the value function as a particular solution to the associated Hamilton–Jacobi–Bellman equation. This approach leads to an implementable numerical method for approximating the value function and optimal reinsurance strategy. Furthermore we give some examples illustrating the applicability of this method for proportional and XL-reinsurance treaties.