Interval oriented multi-section techniques for global optimization

This paper deals with two different optimization techniques to solve the bound-constrained nonlinear optimization problems based on division criteria of a prescribed search region, finite interval arithmetic and interval ranking in the context of a decision maker’s point of view. In the proposed techniques, two different division criteria are introduced where the accepted region is divided into several distinct subregions and in each subregion, the objective function is computed in the form of an interval using interval arithmetic and the subregion containing the best objective value is found by interval ranking. The process is continued until the interval width for each variable in the accepted subregion is negligible. In this way, the global optimal or close to global optimal values of decision variables and the objective function can easily be obtained in the form of an interval with negligible widths. Both the techniques are applied on several benchmark functions and are compared with the existing analytical and heuristic methods.     

Uncertain constrained optimization by interval-oriented algorithm

This paper deals with an interval-oriented approach to solve general interval constrained optimization problems. Generally, this type of problems has infinitely many compromise solutions. The aim of this approach is to obtain one of such solutions with higher accuracy and lower computational cost. The proposed algorithm is nothing but a different kind of branch and bound algorithm with multi-section division criterion of the search region (or box). In the proposed technique, the prescribed/accepted region is divided into several distinct subregions and in each feasible subregion the interval objective function value is computed. Then the subregion containing the best objective value is found by applying a specific interval ranking rule defined with respect to the pessimistic decision makers’ point of view. The process is continued until the interval width for each variable in the accepted subregion is negligible. Finally, the algorithm converges to a compromise solution in interval form. To illustrate the method and also to test the efficiency as well as the effectiveness of the proposed algorithm, we have solved some numerical examples.     

An alternative optimization technique for interval objective constrained optimization problems via multiobjective programming

An alternative optimization technique via multiobjective programming for constrained optimization problems with interval-valued objectives has been proposed. Reduction of interval objective functions to those of noninterval (crisp) one is the main ingredient of the proposed technique. At first, the significance of interval-valued objective functions along with the meaning of interval-valued solutions of the proposed problem has been explained graphically. Generally, the proposed problems have infinitely many compromise solutions. The objective is to obtain one of such solutions with higher accuracy and lower computational effort. Adequate number of numerical examples has been solved in support of this technique.     

区间规划问题的最优性条件

区间规划是带有区间参数的规划问题,是一种更易于求解实际问题的柔性规划。它是确定性优化问题的延伸,有区间线性规划和区间非线性规划两种形式。本文讨论了目标函数是区间函数的区间非线性问题。给出了区间规划问题最优性必要条件的较简单证明方法,并利用LU最优解的概念,在一类广义凸函数－(p,r)－ρ－(η,θ)－不变凸函数定义下讨论了最优性充分条件。     

A mathematical programming approach to sample coefficient of variation with interval-valued observations

The coefficient of variation is a useful statistical measure, which has long been widely used in many areas. In real-world applications, there are situations where the observations are inexact and imprecise in nature and they have to be estimated. This paper investigates the sample coefficient of variation (CV) with uncertain observations, which are represented by interval values. Since the observations are interval-valued, the derived CV should be interval-valued as well. A pair of mathematical programs is formulated to calculate the lower bound and upper bound of the CV. Originally, the pair of mathematical programs is nonlinear fractional programming problems, which do not guarantee to have global optimum solutions. By model reduction and variable substitutions, the mathematical programs are transformed into a pair of quadratic programs. Solving the pair of quadratic programs produces the global optimum solutions and constructs the interval of the CV. The given example shows that the proposed model is indeed able to help the manufacturer select the most suitable manufacturing process with interval-valued observations.     

Necessary Optimality Conditions for Interval Optimization Problems with Functional and Abstract Constraints

Journal of Optimization Theory and Applications - This work addresses interval optimization problems in which the objective function is interval-valued while the constraints are given in functional...     

Research on AHP with interval-valued intuitionistic fuzzy sets and its application in multi-criteria decision making problems

This paper investigates an approach for multi-criterion decision making (MCDM) problems with interval-valued intuitionistic fuzzy preference relations (IVIFPRs). Based on the novel interval score function, some extended concepts associated with IVIFPRs are defined, including the score matrix, the approximate optimal transfer matrix and the possibility degree matrix. By using these new matrixes, a prioritization method for IVIFPRs is proposed. Then, we investigate an interval-valued intuitionistic fuzzy AHP method for multi-criteria decision making (MCDM) problems. In the end, a numerical example is provided to illustrate the application of the proposed approach.     

区间信度环境下基于偏好熵的随机格序排列方法

针对偏好优劣关系的信度为区间值的决策偏好系统,运用熵理论提出了一种基于区间值分布偏好向量的决策分析方法。首先,将决策者对方案的偏好描述由:优于、劣于、等价和不可比这四种关系拓广为优于、劣于、等价、无法比较但有上确界、无法比较但有下确界、无法比较且有上确界又下确界、不可比七种偏好关系,并结合区间证据的概念和性质给出了决策偏好系统的区间值分布偏好向量与相对熵的概念、性质。然后,构建了基于偏好熵的证据推理非线性优化模型,通过求解模型,并结合优先原则和集结规则将个人偏好集结成群体偏好,给出了该决策方法的具体步骤,举例说明了方法的可行性。     

An optimal subgradient algorithm for large-scale bound-constrained convex optimization

This paper shows that the optimal subgradient algorithm (OSGA)—which uses first-order information to solve convex optimization problems with optimal complexity—can be used to efficiently solve arbitrary bound-constrained convex optimization problems. This is done by constructing an explicit method as well as an inexact scheme for solving the bound-constrained rational subproblem required by OSGA. This leads to an efficient implementation of OSGA on large-scale problems in applications arising from signal and image processing, machine learning and statistics. Numerical experiments demonstrate the promising performance of OSGA on such problems. A software package implementing OSGA for bound-constrained convex problems is available.     

Correlation coefficient of interval-valued intuitionistic fuzzy sets and its application to multiple attribute group decision making problems

In this paper, we investigate the group decision making problems in which all the information provided by the decision-makers is presented as interval-valued intuitionistic fuzzy decision matrices where each of the elements is characterized by interval-valued intuitionistic fuzzy number (IVIFN), and the information about attribute weights is partially known. First, we use the interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator to aggregate all individual interval-valued intuitionistic fuzzy decision matrices provided by the decision-makers into the collective interval-valued intuitionistic fuzzy decision matrix, and then we use the score function to calculate the score of each attribute value and construct the score matrix of the collective interval-valued intuitionistic fuzzy decision matrix. From the score matrix and the given attribute weight information, we establish an optimization model to determine the weights of attributes, and then we use the obtained attribute weights and the interval-valued intuitionistic fuzzy weighted geometric (IIFWG) operator to fuse the interval-valued intuitionistic fuzzy information in the collective interval-valued intuitionistic fuzzy decision matrix to get the overall interval-valued intuitionistic fuzzy values of alternatives, and then rank the alternatives according to the correlation coefficients between IVIFNs and select the most desirable one(s). Finally, a numerical example is used to illustrate the applicability of the proposed approach.     

Duality Theory for Optimization Problems with Interval-Valued Objective Functions

A solution concept in optimization problems with interval-valued objective functions, which is essentially similar to the concept of nondominated solution in vector optimization problems, is introduced by imposing a partial ordering on the set of all closed intervals. The interval-valued Lagrangian function and interval-valued Lagrangian dual function are also proposed to formulate the dual problem of the interval-valued optimization problem. Under this setting, weak and strong duality theorems can be obtained.     

Extension of the TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment

TOPSIS is one of the well-known methods for multiple attribute decision making (MADM). In this paper, we extend the TOPSIS method to solve multiple attribute group decision making (MAGDM) problems in interval-valued intuitionistic fuzzy environment in which all the preference information provided by the decision-makers is presented as interval-valued intuitionistic fuzzy decision matrices where each of the elements is characterized by interval-valued intuitionistic fuzzy number (IVIFNs), and the information about attribute weights is partially known. First, we use the interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator to aggregate all individual interval-valued intuitionistic fuzzy decision matrices provided by the decision-makers into the collective interval-valued intuitionistic fuzzy decision matrix, and then we use the score function to calculate the score of each attribute value and construct the score matrix of the collective interval-valued intuitionistic fuzzy decision matrix. From the score matrix and the given attribute weight information, we establish an optimization model to determine the weights of attributes, and construct the weighted collective interval-valued intuitionistic fuzzy decision matrix, and then determine the interval-valued intuitionistic positive-ideal solution and interval-valued intuitionistic negative-ideal solution. Based on different distance definitions, we calculate the relative closeness of each alternative to the interval-valued intuitionistic positive-ideal solution and rank the alternatives according to the relative closeness to the interval-valued intuitionistic positive-ideal solution and select the most desirable one(s). Finally, an example is used to illustrate the applicability of the proposed approach.     

Generalized-Hukuhara penalty method for optimization problem with interval-valued functions and its application in interval-valued portfolio optimization problems

In this study, a gH-penalty method is developed to obtain efficient solutions to constrained optimization problems with interval-valued functions. The algorithmic implementation of the proposed method is illustrated. In order to develop the gH-penalty method, an interval-valued penalty function is defined and the characterization of efficient solutions of a CIOP is done. As an application of the proposed method, a portfolio optimization problem with interval-valued return is solved.     

A Global Optimization Method,QBB, for Twice-Differentiable Nonconvex Optimization Problem

A global optimization method, QBB, for twice-differentiable NLPs (Non-Linear Programming) is developed to operate within a branch-and-bound framework and require the construction of a relaxed convex problem on the basis of the quadratic lower bounding functions for the generic nonconvex structures. Within an exhaustive simplicial division of the constrained region, the rigorous quadratic underestimation function is constructed for the generic nonconvex function structure by virtue of the maximal eigenvalue analysis of the interval Hessian matrix. Each valid lower bound of the NLP problem with the division progress is computed by the convex programming of the relaxed optimization problem obtained by preserving the convex or linear terms, replacing the concave term with linear convex envelope, underestimating the special terms and the generic terms by using their customized tight convex lower bounding functions or the valid quadratic lower bounding functions, respectively. The standard convergence properties of the QBB algorithm for nonconvex global optimization problems are guaranteed. The preliminary computation studies are presented in order to evaluate the algorithmic efficiency of the proposed QBB approach.     

Global optimization and simulated annealing

In this paper we are concerned with global optimization, which can be defined as the problem of finding points on a bounded subset of ⁿ in which some real valued functionf assumes its optimal (maximal or minimal) value.We present a stochastic approach which is based on the simulated annealing algorithm. The approach closely follows the formulation of the simulated annealing algorithm as originally given for discrete optimization problems. The mathematical formulation is extended to continuous optimization problems, and we prove asymptotic convergence to the set of global optima. Furthermore, we discuss an implementation of the algorithm and compare its performance with other well-known algorithms. The performance evaluation is carried out for a standard set of test functions from the literature.     

An efficient algorithm for range computation of polynomials using the Bernstein form

We present a novel optimization algorithm for computing the ranges of multivariate polynomials using the Bernstein polynomial approach. The proposed algorithm incorporates four accelerating devices, namely the cut-off test, the simplified vertex test, the monotonicity test, and the concavity test, and also possess many new features, such as, the generalized matrix method for Bernstein coefficient computation, a new subdivision direction selection rule and a new subdivision point selection rule. The features and capabilities of the proposed algorithm are compared with those of other optimization techniques: interval global optimization, the filled function method, a global optimization method for imprecise problems, and a hybrid approach combining simulated annealing, tabu search and a descent method. The superiority of the proposed method over the latter methods is illustrated by numerical experiments and qualitative comparisons.     

基于区间犹豫模糊信息双向投影技术的双边匹配决策方法

近年来,双边匹配决策问题受到学者们的广泛关注,然而区间犹豫模糊信息双向投影技术的双边匹配决策问题的研究并不多见。因此,针对基于区间犹豫模糊信息下的双边匹配问题,本文提出了一种基于区间犹豫模糊信息双向投影技术的双边匹配决策方法。给出了区间犹豫模糊信息下的双边匹配问题的描述;依据双边主体给出的偏好信息构造区间犹豫模糊矩阵,运用双向投影法对区间犹豫模糊矩阵进行技术处理,构造双向投影矩阵;采用传统Topsis思想构造贴近度矩阵;在此基础上,构建了该双边匹配问题的优化模型,并使用组合满意度法对优化模型进行求解,从而得到了双边主体间的匹配结果;最后,通过算例说明了所提方法的有效性和可行性。     

一类非光滑全局优化问题的区间展开方法

本文利用区间展开的特点,对一类全局优化问题提出一新的区间求解方法,该方 法能处理多元函数的全局优化问题.数值试验表明提出的方法是可行和有效的.     

Global optimization based on novel heuristics,low-discrepancy sequences and genetic algorithms

In this paper a new heuristic hybrid technique for bound-constrained global optimization is proposed. We developed iterative algorithm called GLP_τS that uses genetic algorithms, LP_τ low-discrepancy sequences of points and heuristic rules to find regions of attraction when searching a global minimum of an objective function. Subsequently Nelder–Mead Simplex local search technique is used to refine the solution. The combination of the three techniques (Genetic algorithms, LP_τO Low-discrepancy search and Simplex search) provides a powerful hybrid heuristic optimization method which is tested on a number of benchmark multimodal functions with 10–150 dimensions, and the method properties – applicability, convergence, consistency and stability are discussed in detail.     

一类约束全局优化的区间方法

在区间分析的基础上,对一类不等式约束的全局优化问题,给出几种新的不含全局极小的区域删除准则,提出了一个求不等式约束全局优化问题的区间算法.数值结果表明算法是可行和有效的.