Derivative-Free Methods for Monotone Variational Inequality and Complementarity Problems

Monotone variational inequality problems with box constraints and complementarity problems are reformulated as simple-bound optimization problems. Some derivative-free methods for these problems are proposed. It is shown that, for these new methods, the updated point sequence remains feasible with respect to its simple constraints if the initial point is feasible. Under certain conditions, these methods are globally convergent.     

Mathematical programming via the least-squares method

The least-squares method is used to obtain a stable algorithm for a system of linear inequalities as well as linear and nonlinear programming. For these problems the solution with minimal norm for a system of linear inequalities is found by solving the non-negative least-squares (NNLS) problem. Approximate and exact solutions of these problems are discussed. Attention is mainly paid to finding the initial solution to an LP problem. For this purpose an NNLS problem is formulated, enabling finding the initial solution to the primal or dual problem, which may turn out to be optimal. The presented methods are primarily suitable for ill-conditioned and degenerate problems, as well as for LP problems for which the initial solution is not known. The algorithms are illustrated using some test problems.     

Identification of distributed systems and the theory of the regularization

The identification problems, i.e., the problems of finding unknown parameters in distributed systems from the observations are very important in modern control theory. The solutions of these identification problems can be obtained by solving the equations of the first kind. However, the solutions are often unstable. In other words, they are not continuously dependent on the data. The regularization or Tihonov's regularization is known as one of the stabilizing algorithms to solve these non well-posed problems. In this paper is studied the regularization method for identification of distributed systems. Several approximation theorems are proved to solve the equations of the first kind. Then, identification problems are reduced to the minimization of quadratic cost functionals by virtue of these theorems. On the other hand, it is known that the statistical methods for identification such as the maximum likelihood lead to the minimization problems of certain quadratic functionals. Comparing these quadratic cost functionals, the relations between the regularization and the statistical methods are discussed. Further, numerical examples are given to show the effectiveness of this method.     

Linear programming in tector criterion markov and semi-Markov decision processes

Optimality problems in infinite horizon, discrete time, vector criterion Markov and semi-Markov decision processes are expressed as standard problems of multiobjective linear programming. Processes with discounting, absorbing processes and completely ergodie processes without discounting are investigated. The common properties and special structure of derived multiobjective linear programming problems are overviewed. Computational simplicities associated with these problems in comparison with general multiobjective linear programming problems are discussed. Methods for solving these problems are overviewed and simple numerical examples are given.     

Linear programming for weighted deviation problems using compact basis techniques

Weighted deviation problems are linear programs in which weights (or penalties) are attached to deviations from upper and lower bounds on particular linear expressions. In turn the deviations may be bracketed by secondary bounds. These problems include statistical problems of minimizing weighted sums of absolute deviations, standard and extended “goal programming” problems, problems with upper bounds on absolute values of linear affine functions, problems with arbitrarily bounded variables, and combinations of these.Previous specialized linear programming methods for related problems have been restricted to specialized cases that involve only a single basis configuration, or else, by means of “extended GUB” techniques, accommodate a diverse variety of basis structures at the cost of substantially increased computation. We show that, of the several basis configurations that can arise for this problem, precisely three are essential. Special rules are identified to allow transitions between these three structures, to yield valid compact versions of both the primal and the dual simplex methods. Finally, we show how these results lead to improved efficiency as well as reduced problem size.     

Necessary Optimality Conditions and New Optimization Methods for Cubic Polynomial Optimization Problems with Mixed Variables

Multivariate cubic polynomial optimization problems, as a special case of the general polynomial optimization, have a lot of practical applications in real world. In this paper, some necessary local optimality conditions and some necessary global optimality conditions for cubic polynomial optimization problems with mixed variables are established. Then some local optimization methods, including weakly local optimization methods for general problems with mixed variables and strongly local optimization methods for cubic polynomial optimization problems with mixed variables, are proposed by exploiting these necessary local optimality conditions and necessary global optimality conditions. A global optimization method is proposed for cubic polynomial optimization problems by combining these local optimization methods together with some auxiliary functions. Some numerical examples are also given to illustrate that these approaches are very efficient.     

Differential sensitivity of solutions of convex constrained optimal control problems for discrete systems

A family of optimal control problems for discrete systems that depend on a real parameter is considered. The problems are strongly convex and subject to state and control constraints. Some regularity conditions are imposed on the constraints.The control problems are reformulated as mathematical programming problems. It is shown that both the primal and dual optimal variables for these problems are right-differentiable functions of a parameter. The right-derivatives are characterized as solutions to auxiliary quadratic control problems. Conditions of continuous differentiability are discussed, and some estimates of the rate of convergence of the difference quotients to the respective derivatives are given.     

Experimental study of ILU preconditioners for indefinite matrices

Incomplete LU factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices. However, their failure rate is still too high for them to be useful as black-box library software for general matrices. Besides fatal breakdowns due to zero pivots, the major causes of failure are inaccuracy, and instability of the triangular solves. When there are small pivots, both these problems can occur, but these problems can also occur without small pivots. Through examples from actual problems, this paper shows how these problems evince themselves, how these problems can be detected, and how these problems can sometimes be circumvented through pivoting, reordering, scaling, perturbing diagonal elements, and preserving symmetric structure. The goal of this paper is to gain a better practical understanding of ILU preconditioners and help improve their reliability.     

Fuzzy number linear programming: A probabilistic approach (3)

In the real world there are many linear programming problems where all decision parameters are fuzzy numbers. Several approaches exist which use different ranking functions for solving these problems. Unfortunately when there exist alternative optimal solutions, usually with different fuzzy value of the objective function for these solutions, these methods can not specify a clear approach for choosing a solution. In this paper we propose a method to remove the above shortcoming in solving fuzzy number linear programming problems using the concept of expectation and variance as ranking functions.     

无线通信中的最优资源分配―复杂性分析与算法设计

最优资源分配问题是无线通信系统设计中的基本问题之一.最优地分配功率、传输波形和频谱等资源能够极大地提高整个通信系统的传输性能.目前,相对于通信技术在现实生活中的蓬勃发展,通信系统优化的数学理论和方法显得相对滞后,在某些方面已经成为影响其发展和应用的关键因素.无线通信中的最优资源分配问题常常可建模为带有特殊结构的非凸非线性约束优化问题.一方面,这些优化问题常常具有高度的非线性性,一般情况下难于求解;另一方面,它们又有自身的特殊结构,如隐含的凸性和可分结构等.本文着重考虑多用户干扰信道中物理层资源最优分配问题的复杂性刻画,以及如何利用问题的特殊结构设计有效且满足分布式应用等实际要求的计算方法.     

On solving stochastic production planning problems via scenario modelling

Summary Linear Porgramming models for stochastic planning problems and a methodology for solving them are proposed. A production planning problem with uncertainty in demand is used as a test case, but the methodology presented here is applicable to other types of problems as well. In these models, uncertainty in demand is characterized via scenarios. Solutions are obtained for each scenario and then these individual scenario solutions are aggregated to yield an implementable non-anticipative policy. Such an approach makes it possible to model correlated and nonstationary demand as well as a variety of recourse decision types. For computational purposes, two alternative representations are proposed. A compact approach that is suitable for the Simplex method and a splitting variable approach that is suitable for the Interior Point Methods. A crash procedure that generates an advanced starting solution for the Simplex method is developed. Computational results are reported with both the representations. Although some of the models presented here are very large (over 25000 constraints and 75000 variables), our computational experience with these problems is quite encouraging.     

Optimal Path Planning Based on Visibility

Various problems associated with optimal path planning for mobile observers such as mobile robots equipped with cameras to obtain maximum visual coverage of a surface in the three-dimensional Euclidean space are considered. The existence of solutions to these problems is discussed first. Then, optimality conditions are derived by considering local path perturbations. Numerical algorithms for solving the corresponding approximate problems are proposed. Detailed solutions to the optimal path planning problems for a few examples are given.     

Parallel interior-point solver for structured quadratic programs: Application to financial planning problems

Many practical large-scale optimization problems are not only sparse, but also display some form of block-structure such as primal or dual block angular structure. Often these structures are nested: each block of the coarse top level structure is block-structured itself. Problems with these characteristics appear frequently in stochastic programming but also in other areas such as telecommunication network modelling. We present a linear algebra library tailored for problems with such structure that is used inside an interior point solver for convex quadratic programming problems. Due to its object-oriented design it can be used to exploit virtually any nested block structure arising in practical problems, eliminating the need for highly specialised linear algebra modules needing to be written for every type of problem separately. Through a careful implementation we achieve almost automatic parallelisation of the linear algebra. The efficiency of the approach is illustrated on several problems arising in the financial planning, namely in the asset and liability management. The problems are modelled as multistage decision processes and by nature lead to nested block-structured problems. By taking the variance of the random variables into account the problems become non-separable quadratic programs. A reformulation of the problem is proposed which reduces density of matrices involved and by these means significantly simplifies its solution by an interior point method. The object-oriented parallel solver achieves high efficiency by careful exploitation of the block sparsity of these problems. As a result a problem with over 50 million decision variables is solved in just over 2 hours on a parallel computer with 16 processors. The approach is by nature scalable and the parallel implementation achieves nearly perfect speed-ups on a range of problems. Supported by the Engineering and Physical Sciences Research Council of UK, EPSRC grant GR/R99683/01     

Boundary Value Problems for Degenerating and Nondegenerating Sobolev-Type Equations with a Nonlocal Source in Differential and Difference Forms

Boundary value problems are considered for degenerating and nondegenerating differential equations of the Sobolev type with a nonlocal source as well as finite-difference methods for solving these problems. A priori estimates are derived for solving the problems posed in differential and difference interpretations. These a priori estimates entail the uniqueness and stability of the solution with respect to the initial data and the right-hand side on a layer as well as the convergence of the solution of each difference problem to that of the counterpart differential problem.     

Stability estimates for the solutions to inverse extremal problems for the Helmholtz equation

The inverse problems are under study for the Helmholtz equation describing acoustic scattering at a three-dimensional inclusion. Some optimization method reduces these problems to the inverse extremum problems with variable refraction index and boundary source density as controls. We prove that these problems are solvable and derive the optimality systems that describe necessary optimality conditions. Analysis of the optimality systems leads us to some sufficient conditions on the input data ensuring the uniqueness and stability of optimal solutions.     

A Comparison of Large Scale Mixed Complementarity Problem Solvers

This paper provides a means for comparing various computercodes for solving large scale mixed complementarity problems. Wediscuss inadequacies in how solvers are currently compared, andpresent a testing environment that addresses these inadequacies. Thistesting environment consists of a library of test problems, along withGAMS and MATLAB interfaces that allow these problems to be easilyaccessed. The environment is intended for use as a tool byother researchers to better understand both their algorithms and theirimplementations, and to direct research toward problem classes thatare currently the most challenging. As an initial benchmark, eightdifferent algorithm implementations for large scale mixedcomplementarity problems are briefly described and tested with defaultparameter settings using the new testing environment.     

The stochastic bottleneck linear programming problem

In this paper we consider some stochastic bottleneck linear programming problems. We overview the solution methods in the literature. In the case when the coefficients of the objective functions are simple randomized, the minimum-risk approach will be used for solving these problems. We prove that, under some positivity conditions, these stochastic problems are reduced to certain deterministic bottleneck linear problems. An application of these problems to bottleneck spanning tree problems is given. Two simple numerical examples are presented. This paper was written when I.M. Stancu-Minasian was visiting the Instituto Complutense de Análisis Económico, in the Universidad Complutensen de Madrid, from October 1, 1997 to November 15, 1997 and from October 24, 1998 to November, 9, 1998, as invited researcher. He is grateful to the Institution.     

How to solve a design centering problem

This work considers the problem of design centering. Geometrically, this can be thought of as inscribing one shape in another. Theoretical approaches and reformulations from the literature are reviewed; many of these are inspired by the literature on generalized semi-infinite programming, a generalization of design centering. However, the motivation for this work relates more to engineering applications of robust design. Consequently, the focus is on specific forms of design spaces (inscribed shapes) and the case when the constraints of the problem may be implicitly defined, such as by the solution of a system of differential equations. This causes issues for many existing approaches, and so this work proposes two restriction-based approaches for solving robust design problems that are applicable to engineering problems. Another feasible-point method from the literature is investigated as well. The details of the numerical implementations of all these methods are discussed. The discussion of these implementations in the particular setting of robust design in engineering problems is new.     

Minimizing maximum tardiness and number of tardy jobs on parallel machines subject to minimum flow-time

This paper considers the problems of scheduling jobs on parallel identical machines where an optimal schedule is defined as one that gives the smallest maximum tardiness (or the minimum number of tardy jobs) among the set of schedules with optimal total flow-time (the sum of the completion times of all jobs). We show that these problems are unary NP-Hard, develop lower bounds for these two secondary criteria problems, and describe heuristic algorithms for their solution. Results of a computational study show that the proposed heuristic algorithms are quite effective and efficient in solving these hierarchical criteria scheduling problems.     

Conditions Characterizing Minima of the Difference of Functions

 Optimization problems involving differences of functions arouse interest as generalizations of so-called d.c. problems, i.e. problems involving the difference of two convex functions. The class of d.c. functions is very rich, so d.c. problems are rather general optimization problems. Several global optimality conditions for these d.c. problems have been proposed in the optimization literature. We provide a survey of these conditions and try to detect their common basis. This enables us to give generalizations of the conditions to situations when the objective function is no longer a difference of convex functions, but the difference of two functions which are representable as the upper envelope of an arbitrary family of functions. (Received 6 February 2001; in revised form 11 October 2001)