A Chance Constrained Approach to Fractional Programming with Random Numerator

This paper presents a chance constrained programming approach to the problem of maximizing the ratio of two linear functions of decision variables which are subject to linear inequality constraints. The coefficient parameters of the numerator of the objective function are assumed to be random variables with a known multivariate normal probability distribution. A deterministic equivalent of the stochastic linear fractional programming formulation has been obtained and a subsidiary convex program is given to solve the deterministic problem.     

Numerical algorithm for a class of constrained optimal control problems of switched systems

In this paper, we consider an optimal control problem of switched systems with continuous-time inequality constraints. Because of the complexity of such constraints and switching laws, it is difficult to solve this problem by standard optimization techniques. To overcome the difficulty, we adopt a bi-level algorithm to divide the problem into two nonlinear constrained optimization problems: one continuous and the other discrete. To solve the problem, we transform the inequality constraints into equality constraints which is smoothed using a twice continuously differentiable function and treated as a penalty function. On this basis, the smoothed problem can be solved by any second-order gradient algorithm, e.g., Newton’s Method. Finally, numerical examples show that our method is effective compared to existing algorithms.     

Interval differential evolution with dimension-reduction interval analysis method for uncertain optimization problems

A constrained nonlinear interval optimization method under the framework of differential evolution algorithm is developed to solve the uncertain structural optimization problems with interval uncertainties. The proposed method is a direct optimization method based on the interval differential evolution and dimension-reduction interval analysis. The interval preferential rule based on the satisfaction value of interval possibility degree model is used to realize the direct interval ranking of different design vectors. At each evolutionary generation, the outer optimizer by differential evolution optimizer searches for the best solution within the design space. The dimension-reduction interval analysis is employed to calculate the intervals of objective and constraints for each design vector in the inner layer. This operation transforms the original nesting optimization problem into a single loop one which improves the computational efficiency of the proposed method. Finally, the effectiveness of the presented direct method is verified by two numerical examples and an engineering application.     

基于时间敏感产品的多厂商供应链管理

本文主要研究基于时间敏感产品的多厂商供应链网络模型。在该模型中,每个厂商都希望自己的运营成本和时间消耗最少,但目标函数和约束条件都受到竞争对手决策的影响,因此属于广义纳什均衡问题的范畴。在过去的文献中,这类问题通常被转化成一般形式的变分不等式来处理。本文中,注意到模型中所涉及的约束函数都是线性函数,我们将问题转化成混合互补系统来求解。与一般形式的变分不等式问题相比,混合互补系统要容易处理得多。借助于所谓Fischer-Burmeister函数,我们将混合互补系统转化成非线性方程组,然后利用半光滑牛顿法进行求解。初步的数值实验表明,本文提出的方法是切实可行的。     

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。     

Reliability stochastic optimization for a series system with interval component reliability via genetic algorithm

This paper deals with chance constraints based reliability stochastic optimization problem in the series system. This problem can be formulated as a nonlinear integer programming problem of maximizing the overall system reliability under chance constraints due to resources. The assumption of traditional reliability optimization problem is that the reliability of a component is known as a fixed quantity which lies in the open interval (0, 1). However, in real life situations, the reliability of an individual component may vary due to some realistic factors and it is sensible to treat this as a positive imprecise number and this imprecise number is represented by an interval valued number. In this work, we have formulated the reliability optimization problem as a chance constraints based reliability stochastic optimization problem with interval valued reliabilities of components. Then, the chance constraints of the problem are converted into the equivalent deterministic form. The transformed problem has been formulated as an unconstrained integer programming problem with interval coefficients by Big-M penalty technique. Then to solve this problem, we have developed a real coded genetic algorithm (GA) for integer variables with tournament selection, uniform crossover and one-neighborhood mutation. To illustrate the model two numerical examples have been solved by our developed GA. Finally to study the stability of our developed GA with respect to the different GA parameters, sensitivity analyses have been done graphically.     

Solution of a tropical optimization problem with linear constraints

An optimization problem is considered that is formulated in terms of tropical (idempotent) mathematics and consists in the minimization of a nonlinear function in the presence of linear constraints on the domain of admissible values. The objective function is defined on the set of vectors over an idempotent semifield by a matrix with the use of the operation of multiplicative conjugate transposition. The problem considered is a further generalization of several known problems in which the solution involves the calculation of the spectral radius of the matrix. This generalization implies the use of a more complicated objective function compared with that in the above-mentioned problems, and the imposition of additional constraints. To solve the new problem, an auxiliary variable is introduced that describes the minimum value of the objective function. Then the problem reduces to solving an inequality in which the auxiliary variable plays the role of a parameter. Necessary and sufficient conditions for the existence of solutions to the inequality are used to calculate the parameter, and then the general solution of the inequality is taken as a solution to the original optimization problem. Numerical examples of the solution of problems on the set of two-dimensional vectors are presented.     

A direct solution framework for structural optimization problems with interval uncertainties

A direct solution framework based on multi-objective evolutionary algorithm is developed to solve the structural optimization problems with interval uncertainties. The midpoint and radius of the uncertain original objective are treated as two equally important objectives, which are solved by a multi-objective evolutionary algorithm. The satisfaction value of interval possibility degree model is utilized to deal with nonlinear uncertain constraints and then the degree of constraint violation based on this model is calculated to judge the design vector individuals which one is feasible or infeasible. Subsequently, a selection strategy based on interval constrained-domination rule is utilized to realize the ranking of different design vectors. Finally, two numerical examples and the structural design of augmented reality glasses are investigated to verify the applicability and effectiveness of the proposed method.     

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.     

Numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization

Inspired by the successful applications of the stochastic optimization with second order stochastic dominance (SSD) model in portfolio optimization, we study new numerical methods for a general SSD model where the underlying functions are not necessarily linear. Specifically, we penalize the SSD constraints to the objective under Slater’s constraint qualification and then apply the well known stochastic approximation (SA) method and the level function method to solve the penalized problem. Both methods are iterative: the former requires to calculate an approximate subgradient of the objective function of the penalized problem at each iterate while the latter requires to calculate a subgradient. Under some moderate conditions, we show that w.p.1 the sequence of approximated solutions generated by the SA method converges to an optimal solution of the true problem. As for the level function method, the convergence is deterministic and in some cases we are able to estimate the number of iterations for a given precision. Both methods are applied to portfolio optimization problem where the return functions are not necessarily linear and some numerical test results are reported.     

Direct solution to constrained tropical optimization problems with application to project scheduling

We examine a new optimization problem formulated in the tropical mathematics setting as a further extension of certain known problems. The problem is to minimize a nonlinear objective function, which is defined on vectors over an idempotent semifield by using multiplicative conjugate transposition, subject to inequality constraints. As compared to the known problems, the new one has a more general objective function and additional constraints. We provide a complete solution in an explicit form to the problem by using an approach that introduces an auxiliary variable to represent the values of the objective function, and then reduces the initial problem to a parametrized vector inequality. The minimum of the objective function is evaluated by applying the existence conditions for the solution of this inequality. A complete solution to the problem is given by solving the parametrized inequality, provided the parameter is set to the minimum value. As a consequence, we obtain solutions to new special cases of the general problem. To illustrate the application of the results, we solve a real-world problem drawn from time-constrained project scheduling, and offer a representative numerical example.     

MONOTONIZATION IN GLOBAL OPTIMIZATION

A general monotonization method is proposed for converting a constrained programming problem with non-monotone objective function and monotone constraint functions into a monotone programming problem. An equivalent monotone programming problem with only inequality constraints is obtained via this monotonization method. Then the existing convexification and concavefication methods can be used to convert the monotone programming problem into an equivalent better-structured optimization problem.     

On Extensions of the Frank-Wolfe Theorems

In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty.     

Interactive multiobjective complex search

An interactive procedure based on Box's complex search is used to solve the vector maximization problem. This method has the advantage that the decision maker's underlying value function need not be explicitly specified. Also, the problem may have nonlinear objective functions and nonlinear constraints. Several example problems are presented.     

A cutting plane method for bilevel linear programming with interval coefficients

This article considers the bilevel linear programming problem with interval coefficients in both objective functions. We propose a cutting plane method to solve such a problem. In order to obtain the best and worst optimal solutions, two types of cutting plane methods are developed based on the fact that the best and worst optimal solutions of this kind of problem occur at extreme points of its constraint region. The main idea of the proposed methods is to solve a sequence of linear programming problems with cutting planes that are successively introduced until the best and worst optimal solutions are found. Finally, we extend the two algorithms proposed to compute the best and worst optimal solutions of the general bilevel linear programming problem with interval coefficients in the objective functions as well as in the constraints.     

Multiobjective transportation problem with interval cost,source and destination parameters

In this paper, we focus on the solution procedure of the multiobjective transportation problem (MOTP) where the cost coefficients of the objective functions, and the source and destination parameters have been expressed as interval values by the decision maker. This problem has been transformed into a classical MOTP where to minimize the interval objective function, the order relations that represent the decision maker's preference between interval profits have been defined by the right limit, left limit, centre, and half-width of an interval. The constraints with interval source and destination parameters have been converted into deterministic ones. Finally, the equivalent transformed problem has been solved by fuzzy programming technique. Numerical examples have been provided to illustrate the solution procedure for three possible cases of the original problem.     

A nonlinear programming approach for the sliding mode control design

We treat the sliding mode control problem by formulating it as a two phase problem consisting of reaching and sliding phases. We show that such a problem can be formulated as bicriteria nonlinear programming problem by associating each of these phases with an appropriate objective function and constraints. We then scalarize this problem by taking weighted sum of these objective functions. We show that by solving a sequence of such formulated nonlinear programming problems it is possible to obtain sliding mode controller feedback coefficients which yield a competitive performance throughout the control. We solve the nonlinear programming problems so constructed by using the modified subgradient method which does not require any convexity and differentiability assumptions. We illustrate validity of our approach by generating a sliding mode control input function for stabilization of an inverted pendulum.     

Bounded knapsack sharing

A bounded knapsack sharing problem is a maximin or minimax mathematical programming problem with one or more linear inequality constraints, an objective function composed of single variable continuous functions called tradeoff functions, and lower and upper bounds on the variables. A single constraint problem which can have negative or positive constraint coefficients and any type of continuous tradeoff functions (including multi-modal, multiple-valued and staircase functions) is considered first. Limiting conditions where the optimal value of a variable may be plus or minus infinity are explicitly considered. A preprocessor procedure to transform any single constraint problem to a finite form problem (an optimal feasible solution exists with finite variable values) is developed. Optimality conditions and three algorithms are then developed for the finite form problem. For piecewise linear tradeoff functions, the preprocessor and algorithms are polynomially bounded. The preprocessor is then modified to handle bounded knapsack sharing problems with multiple constraints. An optimality condition and algorithm is developed for the multiple constraint finite form problem. For multiple constraints, the time needed for the multiple constraint finite form algorithm is the time needed to solve a single constraint finite form problem multiplied by the number of constraints. Some multiple constraint problems cannot be transformed to multiple constraint finite form problems.     

Equality Constraints in Multiobjective Robust Design Optimization: Decision Making Problem

Robust design optimization (RDO) problems can generally be formulated by incorporating uncertainty into the corresponding deterministic problems. In this context, a careful formulation of deterministic equality constraints into the robust domain is necessary to avoid infeasible designs under uncertain conditions. The challenge of formulating equality constraints is compounded in multiobjective RDO problems. Modeling the tradeoffs between the mean of the performance and the variation of the performance for each design objective in a multiobjective RDO problem is itself a complex task. A judicious formulation of equality constraints adds to this complexity because additional tradeoffs are introduced between constraint satisfaction under uncertainty and multiobjective performance. Equality constraints under uncertainty in multiobjective problems can therefore pose a complicated decision making problem. In this paper, we provide a new problem formulation that can be used as an effective multiobjective decision making tool, with emphasis on equality constraints. We present two numerical examples to illustrate our theoretical developments.     

一类模糊线性规划模型的模糊最优区间值

讨论一类既有模糊不等式约束又有模糊等式约束的全模糊系数线性规划问题。在给定的模糊隶属度水平下 ,将模型转化为区间数线性规划模型 ,通过确定区间模型的最佳目标函数和最大可行域以及最劣目标函数和最小可行域 ,求出目标函数的模糊最优区间值 ,从而为决策者提供更多的决策信息。最后给出一个数值例子。