A real coded genetic algorithm for solving integer and mixed integer optimization problems

In this paper, a real coded genetic algorithm named MI-LXPM is proposed for solving integer and mixed integer constrained optimization problems. The proposed algorithm is a suitably modified and extended version of the real coded genetic algorithm, LXPM, of Deep and Thakur [K. Deep, M. Thakur, A new crossover operator for real coded genetic algorithms, Applied Mathematics and Computation 188 (2007) 895-912; K. Deep, M. Thakur, A new mutation operator for real coded genetic algorithms, Applied Mathematics and Computation 193 (2007) 211-230]. The algorithm incorporates a special truncation procedure to handle integer restrictions on decision variables along with a parameter free penalty approach for handling constraints. Performance of the algorithm is tested on a set of twenty test problems selected from different sources in literature, and compared with the performance of an earlier application of genetic algorithm and also with random search based algorithm, RST2ANU, incorporating annealing concept. The proposed MI-LXPM outperforms both the algorithms in most of the cases which are considered.     

A Genetic Algorithm for Solving a Capacitated p-Median Problem

Facility-location problems have several applications, such as telecommunications, industrial transportation and distribution. One of the most well-known facility-location problems is the p-median problem. This work addresses an application of the capacitated p-median problem to a real-world problem. We propose a genetic algorithm (GA) to solve the capacitated p-median problem. The proposed GA uses not only conventional genetic operators, but also a new heuristic “hypermutation” operator suggested in this work. The proposed GA is compared with a tabu search algorithm.     

The relax codes for linear minimum cost network flow problems

We describe a relaxation algorithm [1,2] for solving the classical minimum cost network flow problem. Our implementation is compared with mature state-of-the-art primal simplex and primal-dual codes and is found to be several times faster on all types of randomly generated network flow problems. Furthermore, the speed-up factor increases with problem dimension. The codes, called RELAX-II and RELAXT-II, have a facility for efficient reoptimization and sensitivity analysis, and are in the public domain.This work has been supported by the National Science Foundation under Grant NSF-ECS-8217668.     

Computing periodic solutions of linear differential-algebraic equations by waveform relaxation

We propose an algorithm, which is based on the waveform relaxation (WR) approach, to compute the periodic solutions of a linear system described by differential-algebraic equations. For this kind of two-point boundary problems, we derive an analytic expression of the spectral set for the periodic WR operator. We show that the periodic WR algorithm is convergent if the supremum value of the spectral radii for a series of matrices derived from the system is less than 1. Numerical examples, where discrete waveforms are computed with a backward-difference formula, further illustrate the correctness of the theoretical work in this paper.

     

A proximal point method for the sum of maximal monotone operators

In this paper, we concentrate on the maximal inclusion problem of locating the zeros of the sum of maximal monotone operators in the framework of proximal point method. Such problems arise widely in several applied mathematical fields such as signal and image processing. We define two new maximal monotone operators and characterize the solutions of the considered problem via the zeros of the new operators. The maximal monotonicity and resolvent of both of the defined operators are proved and calculated, respectively. The traditional proximal point algorithm can be therefore applied to the considered maximal inclusion problem, and the convergence is ensured. Furthermore, by exploring the relationship between the proposed method and the generalized forward‐backward splitting algorithm, we point out that this algorithm is essentially the proximal point algorithm when the operator corresponding to the forward step is the zero operator. Copyright © 2013 John Wiley & Sons, Ltd.     

Bootstrap Algebraic Multigrid: Status Report,Open Problems,and Outlook

This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies, the least squares method for computing accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in the least squares process. We review some recent research in the development, analysis and application of bootstrap algebraic multigrid and point to open problems in these areas. Results from our previous research as well as some new results for some model diffusion problems with highly oscillatory diffusion coefficient are presented to illustrate the basic components of the BAMG algorithm.     

A rank‐exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem M(λ)x = 0, where M(λ) is a large parameter‐dependent matrix. In several applications, M(λ) has a structure where the higher‐order terms of its Taylor expansion have a particular low‐rank structure. We propose a new Arnoldi‐based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector‐valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. 169–195]. In this paper, the low‐rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way in carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large‐scale problems. Copyright © 2016 John Wiley & Sons, Ltd.     

基于改进区间数密度集结算子指标群赋权方法

针对指标数据已知,而权重数据未知的群组赋权问题,给出了一种基于改进的区间数密度集结算子来进行指标群组赋权的决策方法。首先给出了区间数和区间数密度集结算子(IDM)的定义及性质,改进了以前区间数聚类的方法,应用直接法对一维区间数据组进行聚类,并定义了模糊统计量,以确定最为合理的一种聚类方式。然后基于改进的区间数密度集结算子这种数学模型,来解决指标值数据已知,而权重未知的群组赋权问题。最后举例说明该方法的可行性和实用性。     

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.     

箱覆盖问题的半定松驰算法

箱覆盖问题是NP困难问题中的经典问题，得到了广泛地研究，九十年代以来，半定松驰策略被用来求解组合优化问题，取得了很好的结果[13]，本文首次给箱覆盖问题的半定松驰算法，算法的理论分析结果表明它适合于求解大规模的箱覆盖问题。     

A Hybrid Approximate Extragradient – Proximal Point Algorithm Using the Enlargement of a Maximal Monotone Operator

We propose a modification of the classical extragradient and proximal point algorithms for finding a zero of a maximal monotone operator in a Hilbert space. At each iteration of the method, an approximate extragradient-type step is performed using information obtained from an approximate solution of a proximal point subproblem. The algorithm is of a hybrid type, as it combines steps of the extragradient and proximal methods. Furthermore, the algorithm uses elements in the enlargement (proposed by Burachik, Iusem and Svaiter) of the operator defining the problem. One of the important features of our approach is that it allows significant relaxation of tolerance requirements imposed on the solution of proximal point subproblems. This yields a more practical proximal-algorithm-based framework. Weak global convergence and local linear rate of convergence are established under suitable assumptions. It is further demonstrated that the modified forward-backward splitting algorithm of Tseng falls within the presented general framework.     

A domain decomposition method based on the iterative operator splitting method

In this article a new approach is proposed for constructing a domain decomposition method based on the iterative operator splitting method. The convergence properties of such a method are studied. The main feature of the proposed idea is the decoupling of space and time. We present a multi-iterative operator splitting method that combines iteratively the space and time splitting. We confirm with numerical applications the effectiveness of the proposed iterative operator splitting method in comparison with the classical Schwarz waveform relaxation method as a standard method for domain decomposition. We provide improved results and convergence rates.     

Multigrid Methods with Newton-Gauss-Seidel Smoothing and Constraint Preserving Interpolation for Obstacle Problems

In this paper, we propose a multigrid algorithm based on the full approximate scheme for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A Newton-Gauss-Seidel (NGS) method is used as smoother. A Galerkin coarse grid operator is proposed for the membrane constrained obstacle problem. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. A special prolongation operator is used to interpolate functions accurately from the coarse grid to the fine grid at the boundary between the active and inactive sets. We will demonstrate the fast convergence of the proposed multigrid method for solving two model obstacle problems and compare the results with other multigrid methods.     

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.     

Wave-Shape Function Analysis

We propose to combine cepstrum and nonlinear time–frequency (TF) analysis to study multiple component oscillatory signals with time-varying frequency and amplitude and with time-varying non-sinusoidal oscillatory pattern. The concept of cepstrum is applied to eliminate the wave-shape function influence on the TF analysis, and we propose a new algorithm, named de-shape synchrosqueezing transform (de-shape SST). The mathematical model, adaptive non-harmonic model, is introduced and the de-shape SST algorithm is theoretically analyzed. In addition to simulated signals, several different physiological, musical and biological signals are analyzed to illustrate the proposed algorithm.     

Semidefinite relaxations for semi-infinite polynomial programming

This paper studies how to solve semi-infinite polynomial programming (SIPP) problems by semidefinite relaxation methods. We first recall two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints. Then we propose an exchange algorithm with SDP relaxations to solve SIPP problems with compact index set. At last, we extend the proposed method to SIPP problems with noncompact index set via homogenization. Numerical results show that the algorithm is efficient in practice.     

A MULTI-GRID ALGORITHM FOR MIXED PROBLEMS WITH PENALTY BY C°-PIECEWISE LINEAR ELEMENT APPROXIMATION

In this paper we describe a multi-grid algorithm for mixed problems with penalty by the linear finite element approximation. It is proved that the convergence rate of the algorithm is bound ed away from 1 independently of the meshsize. For convenience, we only discuss Jacobi relaxation as smoothing operator in detail.     

An eigenvalue decomposition based branch-and-bound algorithm for nonconvex quadratic programming problems with convex quadratic constraints

In this paper, we propose a branch-and-bound algorithm for finding a global optimal solution for a nonconvex quadratic program with convex quadratic constraints (NQPCQC). We first reformulate NQPCQC by adding some nonconvex quadratic constraints induced by eigenvectors of negative eigenvalues associated with the nonconvex quadratic objective function to Shor’s semidefinite relaxation. Under the assumption of having a bounded feasible domain, these nonconvex quadratic constraints can be further relaxed into linear ones to form a special semidefinite programming relaxation. Then an efficient branch-and-bound algorithm branching along the eigendirections of negative eigenvalues is designed. The theoretic convergence property and the worst-case complexity of the proposed algorithm are proved. Numerical experiments are conducted on several types of quadratic programs to show the efficiency of the proposed method.     

A genetic algorithm-based heuristic for solving the weighted maximum independent set and some equivalent problems

In this paper we present a genetic algorithm-based heuristic especially for the weighted maximum independent set problem (IS). The proposed approach treats also some equivalent combinatorial optimization problems. We introduce several modifications to the basic genetic algorithm, by (i) using a crossover called two-fusion operator which creates two new different children and (ii) replacing the mutation operator by the heuristic-feasibility operator tailored specifically for the weighted independent set. The performance of our algorithm was evaluated on several randomly generated problem instances for the weighted independent set and on some instances of the DIMACS Workshop for the particular case: the unweighted maximum clique problem. Computational results show that the proposed approach is able to produce high-quality solutions within reasonable computational times. This algorithm is easily parallelizable and this is one of its important features.     

Regularized additive operator-difference schemes

The construction of additive operator-difference (splitting) schemes for the approximate solution Cauchy problem for the first-order evolutionary equation is considered. Unconditionally stable additive schemes are constructed on the basis of the Samarskii regularization principle for operator-difference schemes. In the case of arbitrary multicomponent splitting, these schemes belong to the class of additive full approximation schemes. Regularized additive operator-difference schemes for evolutionary problems are constructed without the assumption that the regularizing operator and the operator of the problem are commutable. Regularized additive schemes with double multiplicative perturbation of the additive terms of the problem’s operator are proposed. The possibility of using factorized multicomponent splitting schemes, which can be used for the approximate solution of steadystate problems (finite difference relaxation schemes) are discussed. Some possibilities of extending the proposed regularized additive schemes to other problems are considered.