The necessary and sufficient conditions for single extremality in non-convex problems of mathematical programming

Mathematical programming is concerned with the extremization of a function f defined over an n-dimensional design space R n and bounded by a set S in the design space. The set S may be defined by equality or inequality constraints, and these constraints may assume linear or nonlinear form. The function f together with the set S in the domain of f is called a mathematical program or a mathematical programming problem. This terminology is in common usage in the context of problems which arise in planning and scheduling which are generally studied under operations research; the branch of mathematics concerned with the decision making process. Mathematical programming problems may be classified into several different categories depending on the nature and form of the design variables, constraint functions, and the objective function. However, only two of these categories are of interest to us, namely the linear and the nonlinear programming (commonly designated as LP and NLP, respectively).KeywordsDesign VariableLinear Programming ProblemCollapse MechanismCollapse LoadBasic Feasible SolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Mathematical programming procedures to optimize the collector field subsystem of a power tower system

This paper establishes a direct method for solving variational problems via a set of Radial basis functions (RBFs) with Gauss-Chebyshev collocation centers. The method consist of reducing a variational problem into a mathematical programming problem. The authors use some optimization techniques to solve the reduced problem. Accuracy and stability of the multiquadric, Gaussian and inverse multiquadric RBF is examined and compared by some numerical experiments.

This paper explores the most urgent task to initialize a virtual network in the network virtualization environment, which always called the virtual network embedding. Different to the previous works to take the embedding process as two steps: node mapping stage and then edge mapping stage, this paper contribute to solve these processes all in one discrete nonlinear combined optimization. Based on modeling the virtual network (VN) request and substrate network (SN), this paper proposed mathematical description to the embedding process which building the relationship between node mapping and edge mapping stages. After that, a virtual network embedding cost is introduced as an objective function, and then the virtual network embedding can be concluding to minimize the objective cost as a mathematical programming problem. The obtained programming results in LINGO9.0 show that the proposed method can finish a virtual network embedding by mathematical programming in one step.

We give an alternative proof of the optimality of the solution for the deterministic EPQ with partial backordering (EPQ-PBO) [Omega, vol. 37, no. 3, pp. 624–636, 2009]. Our proof is based on the mathematical programming theory. We also demonstrate the determination of the optimal decision policy through solving the corresponding mathematical programming problem. We indicate that the same approach can be used within other inventory models with partial backordering, and we consider additional models.

The study deals with the multi-choice mathematical programming problem, where the right hand side of the constraints is multi-choice in nature. However, the problem of multi-choice linear programming cannot be solved directly by standard linear or nonlinear programming techniques. The aim of this paper is to transform such problems to a standard mathematical linear programming problem. For each constraint, exactly one parameter value is selected out of a multiple number of parameter values. This process of selection can be established in different ways. In this paper, we present a new simple technique enabling us to handle such problem as a mixed integer linear programming problem and consequently solve them by using standard linear programming software. Our main aim depends on inserting a specific number of binary variables and using them to construct a linear combination which gives just one parameter among the multiple choice values for each choice of the values of the binary variables. A numerical example is presented to illustrate our analysis.

Chapter 1 - Introduction

Necessary conditions of optimality are obtained for general mathematical programming problems on a product space. The cost functional is locally Lipschitz and the constraints are expressed as inclusion relations with unbounded linear operators and multivalued term. The abstract result is applied to an optimal control problem governed by an elliptic differential inclusion.

The bilevel programming problem (BLPP) is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. To obtain optimality conditions, we reformulate BLPP as a single level mathematical programming problem (SLPP) which involves the value function of the lower level problem. For this mathematical programming problem, it is shown that in general the usual constraint qualifications do not hold and the right constraint qualification is the calmness condition. It is also shown that the linear bilevel programming problem and the minmax problem satisfy the calmness condition automatically. A sufficient condition for the calmness for the bilevel programming problem with quadratic lower level problem and nondegenerate linear complementar¬ity lower level problem are given. First order necessary optimality condition are given using nonsmooth analysis. Second order sufficient optimality conditions are also give...

The ellipsoid yield surface can be applied to many engineering materials, such as metal, concrete, polyester, composite, etc. It is essential to establish an efficient method to carry out the plastic limit analysis of such media. Based on the classical kinematic theorem of plasticity theory and the finite element technique, a discrete mathematical programming formula is presented to determine the collapse load for materials with ellipsoid yield surface. The mathematical programming problem is solved by an efficient algorithm including an iteration process. The convergency of the algorithm is guaranteed. Numerical examples are presented and the results are reasonable and satisfactory.

Convergence with Hilbert's space filling curve

The process of an oil refinery contains both discrete event and continuous variables and can be characterized as a hybrid system. It is extremely challenging to schedule such a system. The short-term scheduling problem of crude oil operations addressed in this chapter is one of the most difficult parts. With jobs to be scheduled being unknown at the beginning, heuristics and meta-heuristics are unable to be applied. Thus, by the existing methods, this problem is formulated as a mathematical programming problem and solved by using exact methods. However, because it is NP-hard in nature, mathematical programming is not computationally efficient and cannot be applied in practice. Up to now, there is no software designed to this problem. In this chapter, for the first time, the problem is studied in a control theory perspective. The problem is modeled by a type of hybrid Petri nets. Based on the model, a two-level control architecture is presented. At the lower level, it solves the schedulability and detailed scheduling problem in a hybrid control theory perspective. At the upper level, it solves a refining scheduling problem, a relative simple problem, with the schedulability conditions as constraints. Consequently, it results in a breakthrough solution such that the large practical application problem can be solved.

An η-approximation approach to duality in mathematical programming problems involving r-invex functions

This paper analyzes the direct and indirect approaches to optimal control via an ordinary gradientrestoration algorithm. First, using the indirect approach, a non-implementable infinite dimensional version of the algorithm, designed to converge to the infinite dimensional necessary conditions of the control problem, is finite-dimensionalized using generic high-order integration schemes. Second, using the direct approach, a finite dimensional version of the algorithm, designed to solve mathematical progranming problems, is applied to the mathematical programming problem which results from the direct transcription of the control problem via the same integration method. By comparing the two algorithms, it is seen how the application of the integration methods to the control problem versus the application of the integration methods to the control problem’s necessary conditions ultimately results in the solution of two different mathematical programming problems. Using an auxillary optimal control problem it is then shown how to implement the infinite dimensional a,gorithm developed using the indirect approach, so that the two finite dimensional algorithms are solving the same problem. This is an important result which unifies the direct and the indirect approaches. It has further significance in that it is now possible, using the indirect approach, to know the exact mathematical programming problem which is being solved, which allows one to accurately compute the Hessian. Exact knowledge of the Hessian is important in preconditioning ill-conditioned problems to enhance the convergence rate of gradient based dgorithms.

A result on Kubatowski's convergence of the intersection of two converging sequences of closed convex subsets of an Euclidean space enables to obtain closedness results for the solution sets of problems with constraints. Such results are applied to minimization problems in mathematical programming, to Nash equilibria in game theory and to competitive equilibria in mathematical economics. Upper semicontinuity results are also derived under usual hypotheses guaranteeing existence of solutions.