Omni-optimizer: A generic evolutionary algorithm for single and multi-objective optimization

Due to the vagaries of optimization problems encountered in practice, users resort to different algorithms for solving different optimization problems. In this paper, we suggest and evaluate an optimization procedure which specializes in solving a wide variety of optimization problems. The proposed algorithm is designed as a generic multi-objective, multi-optima optimizer. Care has been taken while designing the algorithm such that it automatically degenerates to efficient algorithms for solving other simpler optimization problems, such as single-objective uni-optimal problems, single-objective multi-optima problems and multi-objective uni-optimal problems. The efficacy of the proposed algorithm in solving various problems is demonstrated on a number of test problems chosen from the literature. Because of its efficiency in handling different types of problems with equal ease, this algorithm should find increasing use in real-world optimization problems.     

Generalized mixed quasi-equilibrium problems with trifunction

In this paper, we introduce a new class of equilibrium problems, which is called the generalized mixed quasi-equilibrium problems with trifunction. Using the auxiliary principle technique, we suggest and analyze a proximal point method for solving the generalized mixed quasi-equilibrium problems. It is shown that the convergence of the proposed method requires only pseudomonotonicity, which is a weaker condition than monotonicity. Our results represent an improvement and refinement of previously known results. Since the generalized mixed quasi-equilibrium problems include equilibrium problems and variational inequalities as special cases, results proved in this paper continue to hold for these problems.     

Existence of an extremal ground state energy of a nanostructured quantum dot

This paper is concerned with two rearrangement optimization problems. These problems are motivated by two eigenvalue problems which depend nonlinearly on the eigenvalues. We consider a rational and a quadratic eigenvalue problem with Dirichlet’s boundary condition and investigate two related optimization problems where the goal function is the corresponding first eigenvalue. The first eigenvalue in the rational eigenvalue problem represents the ground state energy of a nanostructured quantum dot. In both the problems, the admissible set is a rearrangement class of a given function.     

GASUB: finding global optima to discrete location problems by a genetic-like algorithm

In many discrete location problems, a given number s of facility locations must be selected from a set of m potential locations, so as to optimize a predetermined fitness function. Most of such problems can be formulated as integer linear optimization problems, but the standard optimizers only are able to find one global optimum. We propose a new genetic-like algorithm, GASUB, which is able to find a predetermined number of global optima, if they exist, for a variety of discrete location problems. In this paper, a performance evaluation of GASUB in terms of its effectiveness (for finding optimal solutions) and efficiency (computational cost) is carried out. GASUB is also compared to MSH, a multi-start substitution method widely used for location problems. Computational experiments with three types of discrete location problems show that GASUB obtains better solutions than MSH. Furthermore, the proposed algorithm finds global optima in all tested problems, which is shown by solving those problems by Xpress-MP, an integer linear programing optimizer (21). Results from testing GASUB with a set of known test problems are also provided.     

Variational problems in the theory of optimum processes

This paper presents a review of the optimization problems for control processes described by ordinary differential equations and of the variational methods for solving these problems. The following cases are studied: problems with constraints on the controls or the coordinates, problems described by equations with discontinuous right-hand sides, problems with functionals depending on intermediate coordinates, and problems with given discontinuities in the coordinates. Variational problems of synthesis of optimal systems are also discussed. The method of solution is based on the multiplier rule and the Weierstrass necessary condition for the strong minimum of a functional. In some cases, the Legendre-Clebsch necessary condition for the weak minimum of a functional is used.     

Solving Convex MINLP Optimization Problems Using a Sequential Cutting Plane Algorithm

In this article we look at a new algorithm for solving convex mixed integer nonlinear programming problems. The algorithm uses an integrated approach, where a branch and bound strategy is mixed with solving nonlinear programming problems at each node of the tree. The nonlinear programming problems, at each node, are not solved to optimality, rather one iteration step is taken at each node and then branching is applied. A Sequential Cutting Plane (SCP) algorithm is used for solving the nonlinear programming problems by solving a sequence of linear programming problems. The proposed algorithm generates explicit lower bounds for the nodes in the branch and bound tree, which is a significant improvement over previous algorithms based on QP techniques. Initial numerical results indicate that the described algorithm is a competitive alternative to other existing algorithms for these types of problems.     

基于服装舒适性的纺织材料设计反问题

纺织材料设计反问题是数学物理反问题的一个新领域,也被称为应用数学与计算数学的一个分支.综述纺织材料设计反问题的来源、数学归结,并基于服装的热湿舒适性、压力舒适性提出了设计反问题,给出了反问题解的定义,综述了求解纺织材料设计反问题的数值算法,列举了若干具有挑战的研究课题.     

用非线性方程组求解等式约束非线性规划问题的降维算法

本文研究线性和非线性等式约束非线性规划问题的降维算法.首先,利用一般等式约束问题的降维方法,将线性等式约束非线性规划问题转换成一个非线性方程组,解非线性方程组即得其解;然后,对线性和非线性等式约束非线性规划问题用Lagrange乘子法,将非线性约束部分和目标函数构成增广的Lagrange函数,并保留线性等式约束,这样便得到一个线性等式约束非线性规划序列,从而,又将问题转化为求解只含线性等式约束的非线性规划问题.     

Mathematical Problems of Tomography and Hyperbolic Mappings

We consider a new class of mathematical problems related to interpretation of tomography data. The main assumption is that the sought distribution of absorption is an identically one function in the domain to be determined. These problems are connected with three known directions of mathematical physics: the Dirichlet problems for hyperbolic equations, the problems of small oscillations of a rotating fluid, and the problems of supersonic flows of an ideal gas.     

Helly’s Principle and Its Application to an Infinite-Horizon Optimal Control Problem

In this paper, a numerical method is presented to solve singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a discontinuous source term. First, an asymptotic expansion approximation of the solution of the boundary-value problem is constructed using the basic ideas of the well-known WKB perturbation method. Then, some initial-value problems and terminal-value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial-value problems and terminal-value problems are singularly-perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples are provided to illustrate the method.     

A Branch and Bound Algorithm for Solving Low Rank Linear Multiplicative and Fractional Programming Problems

This paper is concerned with a practical algorithm for solving low rank linear multiplicative programming problems and low rank linear fractional programming problems. The former is the minimization of the sum of the product of two linear functions while the latter is the minimization of the sum of linear fractional functions over a polytope. Both of these problems are nonconvex minimization problems with a lot of practical applications. We will show that these problems can be solved in an efficient manner by adapting a branch and bound algorithm proposed by Androulakis–Maranas–Floudas for nonconvex problems containing products of two variables. Computational experiments show that this algorithm performs much better than other reported algorithms for these class of problems.     

A new constraint test-case generator and the importance of hybrid optimizers

In the past decade, significant progress has been made in understanding problem complexity of discrete constraint problems. In contrast, little similar work has been done for constraint problems in the continuous domain. In this paper, we study the complexity of typical methods for non-linear constraint problems and present hybrid solvers with improved performance. To facilitate the empirical study, we propose a new test-case generator for generating non-linear constraint satisfaction problems (CSPs) and constrained optimization problems (COPs). The optimization methods tested include a sequential quadratic programming (SQP) method, a penalty method with a fixed penalty function, a penalty method with a sequence of penalty functions, and an augmented Lagrangian method. For hybrid solvers, we focus on the form that combines two or more optimization methods in sequence. In the experiments, we apply these methods to solve a series of continuous constraint problems with increasing constraint-to-variable ratios. The test problems include artificial benchmark problems from the test-case generator and problems derived from controlling a hyper-redundant modular manipulator. We obtain novel results on complexity phase transition phenomena of the various methods. Specifically, for constraint satisfaction problems, the SQP method is the best on weakly constrained problems, whereas the augmented Lagrangian method is the best on highly constrained ones. Although the static penalty method performs poorly by itself, by combining it with the SQP method, we show a hybrid solver that is significantly better than any of the individual methods on problems with moderate to large constraint-to-variable ratios. For constrained optimization problems, the hybrid solver obtains much better solutions than SQP, while spending comparable amount of time. In addition, the hybrid solver is flexible and can achieve good results on time-bounded applications by setting parameters according to the time limits.     

Differential evolution framework for big data optimization

During the last two decades, dealing with big data problems has become a major issue for many industries. Although, in recent years, differential evolution has been successful in solving many complex optimization problems, there has been research gaps on using it to solve big data problems. As a real-time big data problem may not be known in advance, determining the appropriate differential evolution operators and parameters to use is a combinatorial optimization problem. Therefore, in this paper, a general differential evolution framework is proposed, in which the most suitable differential evolution algorithm for a problem on hand is adaptively configured. A local search is also employed to increase the exploitation capability of the proposed algorithm. The algorithm is tested on the 2015 big data optimization competition problems (six single objective problems and six multi-objective problems). The results show the superiority of the proposed algorithm to several state-of-the-art algorithms.     

A proof of the Pontryagin maximum principle for initial-value problems

Many optimization problems in economic analysis, when cast as optimal control problems, are initial-value problems, not two-point boundary-value problems. While the proof of Pontryagin (Ref. 1) is valid also for initial-value problems, it is desirable to present the potential practitioner with a simple proof specially constructed for initial-value problems. This paper proves the Pontryagin maximum principle for an initial-value problem with bounded controls, using a construction in which all comparison controls remain feasible. The continuity of the Hamiltonian is an immediate corollary. The same construction is also shown to produce the maximum principle for the problem of Bolza.     

Apply a novel evolutionary algorithm to the solution of parameter selection problems

In this study, a modified line-up competition algorithm (LCA) is used to solve parameter selection problems. The so-called parameter selection problems contain parameter identification problems and optimal control problems. Once the later problems are transformed by control parametrization, the parameters embedded in both problems are selected by the proposed method under the framework of integration approach. Two parameter identification problems and one optimal control problem are given to demonstrate the use of LCA. The results show that in addition to being insensitive to the initial conditions, LCA is very efficient in solving highly nonlinear parameter selection problems.     

一类特殊多项式整数规划问题的最优化算法

本文考虑了一类特殊的多项式整数规划问题。此类问题有很广泛的实际应用,并且是NP难问题。对于这类问题,最优性必要条件和最优性充分条件已经给出。我们在本文中将要利用这些最优性条件设计最优化算法。首 先,利用最优性必要条件,我们给出了一种新的局部优化算法。进而我们结合最优性充分条件、新的局部优化算法和辅助函数,设计了新的全局最优化算法。本文给出的算例展示出我们的算法是有效的和可靠的。     

A simulated annealing hyper-heuristic methodology for flexible decision support

Most of the current search techniques represent approaches that are largely adapted for specific search problems. There are many real-world scenarios where the development of such bespoke systems is entirely appropriate. However, there are other situations where it would be beneficial to have methodologies which are generally applicable to more problems. One of our motivating goals for investigating hyper-heuristic methodologies is to provide a more general search framework that can be easily and automatically employed on a broader range of problems than is currently possible. In this paper, we investigate a simulated annealing hyper-heuristic methodology which operates on a search space of heuristics and which employs a stochastic heuristic selection strategy and a short-term memory. The generality and performance of the proposed algorithm is demonstrated over a large number of benchmark datasets drawn from two very different and difficult problems, namely; course timetabling and bin packing. The contribution of this paper is to present a method which can be readily (and automatically) applied to different problems whilst still being able to produce results on benchmark problems which are competitive with bespoke human designed tailor made algorithms for those problems.     

逆向思维的形式化模型及其应用

逆向思维是一种解决矛盾问题和创新问题的重要思维模式,但目前大部分学者只是运用自然语言对其进行定性研究.利用可拓学的形式化工具——基元和可拓变换,给出了逆向变换和逆向基元的定义,从而构造了逆向思维的形式化模型,这为将来按照一定程序,甚至利用计算机软件生成逆向思维策略解决矛盾问题和创新问题奠定了基础,对于矛盾问题和创新问题的智能化处理研究具有不可替代的意义.     

Complexity and algorithms for convex network optimization and other nonlinear problems

Nonlinear optimization algorithms are rarely discussed from a complexity point of view. Even the concept of solving nonlinear problems on digital computers is not well defined. The focus here is on a complexity approach for designing and analyzing algorithms for nonlinear optimization problems providing optimal solutions with prespecified accuracy in the solution space. We delineate the complexity status of convex problems over network constraints, dual of flow constraints, dual of multi-commodity, constraints defined by a submodular rank function (a generalized allocation problem), tree networks, diagonal dominant matrices, and nonlinear Knapsack problem's constraint. All these problems, except for the latter in integers, have polynomial time algorithms which may be viewed within a unifying framework of a proximity-scaling technique or a threshold technique. The complexity of many of these algorithms is furthermore best possible in that it matches lower bounds on the complexity of the respective problems. In general nonseparable optimization problems are shown to be considerably more difficult than separable problems. We compare the complexity of continuous versus discrete nonlinear problems and list some major open problems in the area of nonlinear optimization. MSC classification: 90C30, 68Q25     

Parallel optimization for traffic assignment

Most large-scale optimization problems exhibit structures that allow the possibility of attack via algorithms that exhibit a high level of parallelism. The emphasis of this paper is the development of parallel optimization algorithms for a class of convex, block-structured problems. Computational experience is cited for some large-scale problems arising from traffic assignment applications. The algorithms considered here have the property that they allow such problems to be decomposed into a set of smaller optimization problems at each major iteration. These smaller problems correspond to linear single-commodity networks in the traffic assignment case, and they may be solved in parallel. Results are given for the distributed solution of such problems on the CRYSTAL multicomputer.This research was supported in part by NSF grant CCR-8709952 and AFOSR grant AFOSR-86-0194.