Zero-one Programming Problem Research Articles

On choosing initial values of iteratively reweighted ℓ1 algorithms for the piece-wise exponential penalty

Computing the proximal operator of the sparsity-promoting piece-wise exponential (PiE) penalty [Formula: see text] with a given shape parameter [Formula: see text], which is treated as a popular nonconvex surrogate of [Formula: see text]-norm, is fundamental in feature selection via support vector machines, image reconstruction, zero-one programming problems, compressed sensing, neural networks, etc. Due to the nonconvexity of PiE, for a long time, its proximal operator is frequently evaluated via an iteratively reweighted [Formula: see text] algorithm, which substitutes PiE with its first-order approximation, however, the obtained solutions only are the critical point. Based on the exact characterization of the proximal operator of PiE, we explore how the iteratively reweighted [Formula: see text] solution deviates from the true proximal operator in certain regions, which can be explicitly identified in terms of [Formula: see text], the initial value and the regularization parameter in the definition of the proximal operator. Moreover, the initial value can be adaptively and simply chosen to ensure that the iteratively reweighted [Formula: see text] solution belongs to the proximal operator of PiE.

Solving the capacitated vehicle routing problem with Lexi-search approach

In this paper we present a variant vehicle routing problem called “Solving the Capacitated Vehicle Routing Problem with Lexi-Search Approach” (CVRP). The purpose of this article is to propose an efficient Lexi-Search Algorithm using pattern recognition technique for solving CVRP on a scalable multicomputer platform and to obtain an optimal solution. Our results show that the proposed algorithm is highly competitive on a set of benchmark problems. In this paper we focus our investigation on solving the capacitated VPR (CVPR) and considered a variant vehicle routing problem called as “ Solving the Capacitated Vehicle Routing Problem with Lexi-Search Approach”. First the model is formulated into a zero-one programming problem. A Lexi-Search Algorithm using Pattern Recognition Technique is developed for getting an optimal solution. The problem is discussed with suitable numerical illustration. We have programmed the proposed algorithm using C-language. The computational details are reported. As an observation the CPU run time is fairly less for higher values to the parameters of the problem to obtain optimal solutions.

Solving the capacitated vehicle routing problem with Lexi-search approach

In this paper we present a variant vehicle routing problem called “Solving the Capacitated Vehicle Routing Problem with Lexi-Search Approach” (CVRP). The purpose of this article is to propose an efficient Lexi-Search Algorithm using pattern recognition technique for solving CVRP on a scalable multicomputer platform and to obtain an optimal solution. Our results show that the proposed algorithm is highly competitive on a set of benchmark problems. In this paper we focus our investigation on solving the capacitated VPR (CVPR) and considered a variant vehicle routing problem called as “Solving the Capacitated Vehicle Routing Problem with Lexi-Search Approach”. First the model is formulated into a zero-one programming problem. A Lexi-Search Algorithm using Pattern Recognition Technique is developed for getting an optimal solution. The problem is discussed with suitable numerical illustration. We have programmed the proposed algorithm using C-language. The computational details are reported. As an observation the CPU run time is fairly less for higher values to the parameters of the problem to obtain optimal solutions.

Discrete self-organizing migration algorithm and p-location problems

Mathematical modelling, and integer programming generally, has many practical applications in different areas of human life. Effective and fast solving approaches for various optimization problems play an important role in the decision-making process and therefore, big attention is paid to the development of many exact and approximate algorithms. This paper deals only with a special class of location problems in which given number of facilities are to be chosen to minimize the objective function value. Since the exact methods are not suitable for their unpredictable computational time or memory demands, we focus here on possible usage of a special type of a particle swarm optimization algorithm transformed by discretization and meme usage into so-called discrete self-organizing migrating algorithm. In the paper, there is confirmed that it is possible to suggest a sophisticated heuristic for zero-one programming problem, which can produce near-to-optimal solution in much smaller time than the time demanded by exact methods. We introduce a special adaptation of the discrete self-organizing migration algorithm to the $p$-location problem making use of the path-relinking method. In the theoretical part of this paper, we introduce several strategies of the migration process. To verify their features and effectiveness, a computational study with real-sized benchmarks was performed. The main goal of the experiments was to find the most efficient version of the suggested solving tool.

An artificial neural network for solving quadratic zero-one programming problems

This paper presents an artificial neural network to solve the quadratic zero-one programming problems under linear constraints. In this paper, by using the connection between integer and nonlinear programming, the quadratic zero-one programming problem is transformed into the quadratic programming problem with nonlinear constraints. Then, by using the nonlinear complementarity problem (NCP) function and penalty method this problem is transformed into an unconstrained optimization problem. It is shown that the Hessian matrix of the associated function in the unconstrained optimization problem is positive definite in the optimal point. To solve the unconstrained optimization problem an artificial neural network is used. The proposed neural network has a simple structure and a low complexity of implementation. It is shown here that the proposed artificial neural network is stable in the sense of Lyapunov. Finally, some numerical examples are given to show that the proposed model finds the optimal solution of this problem in the low convergence time.

Symmetry in RLT-type relaxations for the quadratic assignment and standard quadratic optimization problems

The reformulation–linearization technique (RLT), introduced in [Sherali, H. D., Adams. W. P. (1990). A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3(3), 411–430], provides a way to compute a hierarchy of linear programming bounds on the optimal values of NP-hard combinatorial optimization problems. In this paper we show that, in the presence of suitable algebraic symmetry in the original problem data, it is sometimes possible to compute level two RLT bounds with additional linear matrix inequality constraints. As an illustration of our methodology, we compute the best-known bounds for certain graph partitioning problems on strongly regular graphs.

A note on solving MINLP’s using formulation space search

In this note we present an approach based on formulation space search to solve mixed-integer nonlinear (zero-one) programming problems. Our approach is an iterative one which adds a single nonlinear inequality constraint of increasing tightness to the original problem. Computational results are presented for our approach on 51 standard benchmark problems taken from MINLPLib. We compare our approach against the Minotaur and minlp_bb nonlinear solvers, as well as against the RECIPE algorithms.

A Systematic Method for Constructing Feasible Solution to SCUC Problem With Analytical Feasibility Conditions

Obtaining high-quality feasible solution is the core and the major difficulty in solving security-constrained unit commitment (SCUC) problems. This paper presents a systematic method for constructing feasible solutions to SCUC problem based on a group of analytical feasibility conditions. The feasibility check is performed based on the analytical necessary conditions such that most of infeasible UC states can be identified without solving LP problem. If a UC state is infeasible, it is adjusted with the possibly minimal operating cost increase based on the cost information. This UC adjusting issue is formulated as a zero-one programming problem and a branch and bound (B&B) method is established based on these feasibility conditions. Numerical testing is performed for a 31-bus system, an IEEE 24-bus system, and an IEEE 118-bus system. The testing results suggest that over 95% of infeasible UC states are identified by the analytical necessary conditions. The near-optimal feasible schedules for SCUC problem can be obtained efficiently by the proposed method. The feasible schedules obtained are compared with those obtained from mixed integer programming-based method in the IEEE 118-bus system. It is shown that the new method can produce competitive results in terms of solution quality and computational efficiency.

Conflict graphs in implicit enumeration

A zero-one linear program is a global discrete optimization problem. Namely, it is a linear programming problem with zero-one variables. The real relaxation of a zero-one program is again a continuous linear programming problem. Simply, the zero-one variables are replaced by continuous variables varying between zero and one independently of each other. If the real relaxation, as a continuous problem, solved with the simplex method happens to have a zero-one optimal solution, then this particular solution is also an optimal solution of the original zero-one programming problem. Suppose that x1,…,xn are all the variables of a given zero-one linear programming problem. For convenience we introduce further variables y1,…,yn defined by y1=1−x1,…, yn=1−xn respectively. For a unified notation the variables x1,…,xn and y1,…, yn together will be denoted by z1,…,z2n. These variables will play the roles the nodes of the conflict graph Г associated with the given zero-one linear programming problem. Fixing the variables zi and zj both to be equal to one simultaneously reduces the number of variables in the problem. If the new smaller optimization problem does not have any feasible solution, then the nodes zi and zj will be connected by an edge in the graph Γ. The graph Γ simply records that there is a conflict between the assignments zi =1 and zj =1 which explains the name conflict graph. In other words the non-directed edge between the nodes zi and zj codes the fact that the inequality zi+zj≤1holds. The conflict graph was designed to generate additional constraints the so-called valid inequalities or cuts in order to expedite the solution of the zero-one linear program via its real relaxation. This particular solution strategy is aptly named the branch and cut method. It will be shown that the conflict graph besides its intended use can also be applied in another solution technique the method of implicit enumeration. We will illustrate by examples that the conflict graph provides us with rules to prune the search tree that are not offered by the commonly applied pruning rules. In addition the computations involved can easily be organized into a highly parallel computational scheme.

Convergence of a continuous approach for zero-one programming problems

In this paper a new continuous formulation for the zero-one programming problem is presented, followed by an investigation of the algorithm for it. This paper first reformulates the zero-one programming problem as an equivalent mathematical programs with complementarity constraints, then as a smooth ordinary nonlinear programming problem with the help of the Fischer–Burmeister function. After that the augmented Lagrangian method is introduced to solve the resulting continuous problem, with optimality conditions for the non-smooth augmented Lagrangian problem derived on the basis of approximate smooth variational principle, and with convergence properties established. To our benefit, the sequence of solutions generated converges to feasible solutions of the original problem, which provides a necessary basis for the convergence results.

A gradient-based approach for discrete optimum design

In this paper, structural optimization with discrete variables in engineering design is modeled and investigated as a zero-one programming problem. The zero-one programming problem is first reformulated as an equivalent continuous problem through replacing the zero-one constraints by complementarity constraints, then as an equivalent ordinary nonlinear programming problem with the help of the NCP(nonlinear complementarity problem) function. Furthermore, an aggregate function method is introduced with the aim to simplify computation in the following augmented Lagrangian method. Numerical experiments on discrete optimum design showed the proposed method is promising.

Stochastic zero – one programming

zero - one programming case from integer linear programming where the variable's are equal to zero or one, the decision factor uses this kind from programming when he meets him problems of the kind yes or no. The stochastic zero-one programming construction formed is used when one or all parameters of model(cj,bi,aij) are random variable taken mathematical distribution. In this research we discuss stochastic zero-one programming problem where (aij) random variable (construction and solution) and use it in practical application on some vegetative crops in Iraq.

New Implicit Enumeration Method for Polynomial 0-1 Programming

Abstract A new implicit enumeration method for polynomial zero-one programming is proposed in this article. By adopting the p-norm surrogate constraint method, a polynomial zero-one programming problem with multiple constraints can be converted into an equivalent polynomial zero-one programming problem with a single surrogate constraint. A new solution scheme is then devised to take the advantage of this prominent feature in carrying out the “fathoming” procedure and the “backtrack” procedure in a searching process of an implicit enumeration. We demonstrate the efficiency of this new algorithm by some promising computational results. Finally, we conclude by proposing certain topics for future research.

COMPUTATIONAL METHODS THROUGH GENETIC ALGORITHMS FOR OBTAINING STACKELBERG SOLUTIONS TO TWO-LEVEL MIXED ZERO-ONE PROGRAMMING PROBLEMS

In this paper, we develop computational methods for obtaining Stackelberg solutions to two-level mixed zero-one programming problems in which the decision maker at the upper level controls zero-one variables and the decision maker at the lower level controls real variables. To illustrate two-level mixed zero-one programming problems, we formulate a facility location and transportation problem as a two-level mixed zero-one programming problem. We develop computational methods through genetic algorithms for obtaining Stackelberg solutions. To demonstrate the feasibility and efficiency of the proposed methods, computational experiments are carried out and comparisons between the methods based on the branch-and-bound techniques and the proposed methods are provided.

分権的2レベル0-1計画問題に対する遺伝的アルゴリズムを用いたStackelberg-Nash解の計算方法

分権的2レベル0-1計画問題に対する遺伝的アルゴリズムを用いたStackelberg-Nash解の計算方法

2レベル混合0-1計画問題に対する遺伝的アルゴリズムを用いたStackelberg解の計算方法

本論文では, 上位レベルの意思決定者の決定変数が0-1変数で下位レベルの意思決定者の決定変数が非負の連続変数であり, それぞれの意思決定者は互いに異なる目的関数を最小化する2レベル線形計画問題を取り扱う.2レベルの問題において, 上位レベルの意思決定者が先に決定し, 下位レベルの意思決定者がそれに続いて決定する場合の非協力状況下でのStackelberg解を求めるために, 遺伝的アルゴリズムに基づいた計算方法を提案する.数値実験を行い, 提案する手法の有効性と実行可能性を示すと同時に, 従来の解法との比較を行う.

A general form for the capital projects sequencing problem

The capital sequencing problem consists of finding the lowest cost starting times for a set of capital projects when such projects: 1) are exposed to the effects of inflation, 2) cost different amounts, 3) require different lengths of time to complete, and 4) have dissimilar cash outflows per period. The problem is non-trivial when the timing of the projects is constrained by any one, or any combination of, twelve possibilities. The problem is modeled as a straightforward mathematical zero-one programming problem, and is complicated only by the large number of constraints which are typically needed for a problem of realistic size.

Using Simulated Annealing Heuristic to the Lot Sizing with Overtime Production.

A heuristic based on simulated annealing method is developed to solve lot sizing problem with setup times. The problem assumes that the unlimited overtime production is available for each planning period so as to generate a feasible neighborhood solution easily. Firstly the problem is modified into a zero-one programming problem in order to obtain the objective function value without complicated calculation. Then the simulated annealing heuristic is implemented to obtain the best solution. Finally this solution is fed into a simplex algorithm to obtain the revised best solution. In the numerical examples, the simulated annealing heuristic is compared with the Lagrangean relaxation approach and the following results are obtained ; ( 1 ) the heuristic produced better solutions on the average, ( 2 ) setting the production quantity equal to either zero or the cumulative demand performed well in the condition of many products, and ( 3 ) performance of the heuristic was relatively sensitive to the initial random seeds.

A Simulated Annealing Algorithm for General Zero-One Programming Problems

Abstract We present a way to incorporate a simulated annealing approach when a general zero-one programming problem is solved by means of linear programming and classical branch and bound. The method is based on a repetitive application of the Pivot and Complement heuristic in which cutting planes are added to avoid repetition of previously obtained solutions and where an objective function cut is added in case the solution obtained is regarded as inappropriate in terms of the simulated annealing idea. The method is illustrated on a set of multiconstraint knapsack problems.

A Hierarchy of Relaxations between the Continuous and Convex Hull Representations for Zero-One Programming Problems

In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into an extended linear program. It is shown that the strength of the resulting reformulation depends on the degree of the terms used to produce the polynomial program at the intermediate step of this method. In fact, as this degree varies from one up to the number of variables in the problem, a hierarchy of sharper representations is obtained with the final relaxation representing the convex hull of feasible solutions. The reformulation technique readily extends to produce a similar hierarchy of linear relaxations for zero-one polynomial programming problems. A characterization of the convex hull in the original variable space is also available through a projection process. The structure of this convex hull characterization (or its other relaxations) can be exploited to generate strong or facetial valid inequalities through appropriate surrogates in a computational framework. The surrogation process can also be used to study various classes of facets for different combinatorial optimization problems. Some examples are given to illustrate this point.