On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.     

Possibility and necessity representations of fuzzy inequality and its application to two warehouse production-inventory problem

In this paper, possibility and necessity representations of fuzzy inequality constraints are presented and then crisp versions of the constraints are derived. Here analogous to chance constraints, real-life necessity and possibility constraints in the context of two warehouse multi-item dynamic production-inventory control system are defined and defuzzified following fuzzy relations. Hence, a realistic two warehouse multi-item production-inventory model with fuzzy constraints has been formulated for a finite period of time and solved for optimal production with the objective of having maximum profit. The rate of production is unknown, assumed to be a function of time and considered as a control variable. Also the present system produces some defective units alongwith the perfect ones and the rate of produced defective units is stochastic in nature. Demand of the good units is stock dependent and known and the defective units are sold at a reduced price. The space required per unit item and available storage space are assumed to be imprecise. The inequality of budget constraints is also imprecise. The space and budget constraints are expressed as necessity and/or possibility types. The model is reduced to an equivalent deterministic model using fuzzy relations and solved for optimum production function using Pontryagin’s optimal control policy, the Kuhn–Tucker conditions and generalized reduced gradient (GRG) technique. The model is illustrated numerically and values of demand, optimal production function and stock level are presented in both tabular and pictorial forms.     

Abstract convex approximations of nonsmooth functions

In the article we use abstract convexity theory in order to unify and generalize many different concepts of nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality conditions for nonsmooth nonconvex optimization problems with the abstract codifferentiable or abstract quasidifferentiable objective function and constraints. Then, we demonstrate how these conditions can be transformed into simpler and more constructive conditions in some particular cases.     

Smoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraints

The equilibrium problem with equilibrium constraints (EPEC) can be looked on as a generalization of Nash equilibrium problem (NEP) and the mathematical program with equilibrium constraints (MPEC) whose constraints contain a parametric variational inequality or complementarity system. In this paper, we particularly consider a special class of EPECs where a common parametric P-matrix linear complementarity system is contained in all players?? strategy sets. After reformulating the EPEC as an equivalent nonsmooth NEP, we use a smoothing method to construct a sequence of smoothed NEPs that approximate the original problem. We consider two solution concepts, global Nash equilibrium and stationary Nash equilibrium, and establish some results about the convergence of approximate Nash equilibria. Moreover we show some illustrative numerical examples.     

Minimal infeasible constraint sets in convex integer programs

In this paper we investigate certain aspects of infeasibility in convex integer programs, where the constraint functions are defined either as a composition of a convex increasing function with a convex integer valued function of n variables or the sum of similar functions. In particular we are concerned with the problem of an upper bound for the minimal cardinality of the irreducible infeasible subset of constraints defining the model. We prove that for the considered class of functions, every infeasible system of inequality constraints in the convex integer program contains an inconsistent subsystem of cardinality not greater than 2 ⁿ , this way generalizing the well known theorem of Scarf and Bell for linear systems. The latter result allows us to demonstrate that if the considered convex integer problem is bounded below, then there exists a subset of at most 2 ⁿ −1 constraints in the system, such that the minimum of the objective function subject to the inequalities in the reduced subsystem, equals to the minimum of the objective function over the entire system of constraints.     

On Quantitative Stability in Optimization and Optimal Control

We study two continuity concepts for set-valued maps that play central roles in quantitative stability analysis of optimization problems: Aubin continuity and Lipschitzian localization. We show that various inverse function theorems involving these concepts can be deduced from a single general result on existence of solutions to an inclusion in metric spaces. As applications, we analyze the stability with respect to canonical perturbations of a mathematical program in a Hilbert space and an optimal control problem with inequality control constraints. For stationary points of these problems, Aubin continuity and Lipschitzian localization coincide; moreover, both properties are equivalent to surjectivity of the map of the gradients of the active constraints combined with a strong second-order sufficient optimality condition.     

Optimizing Preventive Maintenance Models

We deal with the problem of scheduling preventive maintenance (PM) for a system so that, over its operating life, we minimize a performance function which reflects repair and replacement costs as well as the costs of the PM itself. It is assumed that a hazard rate model is known which predicts the frequency of system failure as a function of age. It is also assumed that each PM produces a step reduction in the effective age of the system. We consider some variations and extensions of a PM scheduling approach proposed by Lin et al. [6]. In particular we consider numerical algorithms which may be more appropriate for hazard rate models which are less simple than those used in [6] and we introduce some constraints into the problem in order to avoid the possibility of spurious solutions. We also discuss the use of automatic differentiation (AD) as a convenient tool for computing the gradients and Hessians that are needed by numerical optimization methods. The main contribution of the paper is a new problem formulation which allows the optimal number of occurrences of PM to be determined along with their optimal timings. This formulation involves the global minimization of a non-smooth performance function. In our numerical tests this is done via the algorithm DIRECT proposed by Jones et al. [19]. We show results for a number of examples, involving different hazard rate models, to give an indication of how PM schedules can vary in response to changes in relative costs of maintenance, repair and replacement. Part of this work was carried out while the first author was a Visiting Professor in the Department of Mechanical Engineering at the University of Alberta in December 2003.     

Possibility and necessity constraints and their defuzzification—A multi-item production-inventory scenario via optimal control theory

In this paper, analogous to chance constraints, real-life necessity and possibility constraints in the context of a multi-item dynamic production-inventory control system are defined and defuzzified following fuzzy relations. Hence, a realistic multi-item production-inventory model with shortages and fuzzy constraints has been formulated and solved for optimal production with the objective of having minimum cost. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the present system produces some defective units along with the perfect ones and the rate of produced defective units is constant. Here demand of the good units is time dependent and known and the defective units are of no use. The space required per unit item, available storage space and investment capital are assumed to be imprecise. The space and budget constraints are of necessity and/or possibility types. The model is formulated as an optimal control problem and solved for optimum production function using Pontryagin’s optimal control policy, the Kuhn–Tucker conditions and generalized reduced gradient (GRG) technique. The model is illustrated numerically and values of demand, optimal production function and stock level are presented in both tabular and graphical forms. The sensitivity of the cost functional due to the changes in confidence level of imprecise constraints is also presented.     

Superiority–inferiority modeling coupled minimax-regret analysis for energy management systems

In this study, a superiority–inferiority-based minimax-regret analysis (SI-MRA) model is developed for supporting the energy management systems (EMS) planning under uncertainty. In SI-MRA model, techniques of fuzzy mathematical programming (FMP) with the superiority and inferiority measures and minimax regret analysis (MMR) are incorporated within a general framework. The SI-MRA improves upon conventional FMP methods by directly reflecting the relationships among fuzzy coefficients in both the objective function and constraints with a high computational efficiency. It can not only address uncertainties expressed as fuzzy sets in both of the objective function and system constraints but also can adopt a list of scenarios to reflect the uncertainties of random variables without making assumptions on their possibilistic distributions. The developed SI-MRA model is applied to a case study of long-term EMS planning, where fuzziness and randomness exist in the costs for electricity generation and demand. A number of scenarios associated with various alternatives and outcomes under different electricity demand levels are examined. The results can help decision makers identify an optimal strategy of planning electricity generation and capacity expansion based on a minimax regret level under uncertainty.     

A mathematical programming model and solution for scheduling production orders in Shanghai Baoshan Iron and Steel Complex

In this paper, we investigate the production order scheduling problem derived from the production of steel sheets in Shanghai Baoshan Iron and Steel Complex (Baosteel). A deterministic mixed integer programming (MIP) model for scheduling production orders on some critical and bottleneck operations in Baosteel is presented in which practical technological constraints have been considered. The objective is to determine the starting and ending times of production orders on corresponding operations under capacity constraints for minimizing the sum of weighted completion times of all orders. Due to large numbers of variables and constraints in the model, a decomposition solution methodology based on a synergistic combination of Lagrangian relaxation, linear programming and heuristics is developed. Unlike the commonly used method of relaxing capacity constraints, this methodology alternatively relaxes constraints coupling integer variables with continuous variables which are introduced to the objective function by Lagrangian multipliers. The Lagrangian relaxed problem can be decomposed into two sub-problems by separating continuous variables from integer ones. The sub-problem that relates to continuous variables is a linear programming problem which can be solved using standard software package OSL, while the other sub-problem is an integer programming problem which can be solved optimally by further decomposition. The subgradient optimization method is used to update Lagrangian multipliers. A production order scheduling simulation system for Baosteel is developed by embedding the above Lagrangian heuristics. Computational results for problems with up to 100 orders show that the proposed Lagrangian relaxation method is stable and can find good solutions within a reasonable time.     

A multi-objective production smoothing model with compressible operating times

Real life multi-product multi-period production planning often deals with several conflicting objectives while considering a set of technological constraints. The solutions of these problems can provide deeper insights to the decision makers/managers than those of single-objective problems. Some managers want to use from a production plan that is corresponding to minimum change in production policy along with minimum total cost simultaneously as possible. On the other hand, these two objectives have intrinsic conflicts such that producing in a fixed rate will cause huge costs than producing economically or according to JIT. So this paper presents a novel multi-objective model for the production smoothing problem on a single stage facility that some of the operating times could be determined in a time interval for. The model is to: (a) smooth the variations of production volume, and (b) minimize total cost of the corresponding production plan, while satisfying a set of technological constraints such as limited available time. The proposed model is developed in a real case study and is solved by a new genetic algorithm. The proposed genetic algorithm uses a novel achievement function for exploring the solution space, based on LP-metric concepts. Computational experiences reveal the sufficiency and efficiency of the proposed approach in contrast to previous researches.     

Canonical duality for solving general nonconvex constrained problems

This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem (without duality gap) can be obtained in a unified form with global optimality conditions provided.While the popular augmented Lagrangian method may produce more difficult nonconvex problems due to the nonlinearity of constraints. Some fundamental concepts such as the objectivity and Lagrangian in nonlinear programming are addressed.     

On the attainability problem under state constraints with piecewise smooth boundary

The paper is devoted to the problem of approximating reachable sets for a nonlinear control system with state constraints given as a solution set of a finite system of nonlinear inequalities. Each of these inequalities is given as a level set of a smooth function, but their intersection may have nonsmooth boundary. We study a procedure of eliminating the state constraints based on the introduction of an auxiliary system without constraints such that the right-hand sides of its equations depend on a small parameter. For state constraints with smooth boundary, it was shown earlier that the reachable set of the original system can be approximated in the Hausdorff metric by the reachable sets of the auxiliary control system as the small parameter tends to zero. In the present paper, these results are extended to the considered class of systems with piecewise smooth boundary of the state constraints.     

An RQP algorithm using a differentiable exact penalty function for inequality constrained problems

In this paper we propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function. The algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an Armijo-type line search procedure that avoids the need to evaluate second order derivatives of the problem functions. We prove that the algorithm possesses global and superlinear convergence properties. Numerical results are reported.     

A neural network system for solving an assortment problem in the steel industry

A neural network model for solving an assortment problem found in the iron and steel industry is discussed in this paper. The problem arises in the yard where steel plate is cut into rectangular pieces. The neural network model can be categorized as a Hopfield model, but the model is expanded to handle inequality constraints. The idea of a penalty function is used. A large penalty is applied to the network if a constraint is not satisfied. The weights are updated based on the penalty values. A special term is added to the energy function of the network to guarantee the convergence of the neural network which has this feature. The performance of the neural network was evaluated by comparison with an existing expert system. The results showed that the neural network has the potential to identify in a short time near-optimal solutions to the assortment problem. The neural network is used as the core of a system for dealing with the assortment problem. In building the neural networks system for practical use, there were many implementation issues. Some of them are presented here, and the fundamental ideas are explained. The performance of the neural network system is compared to that of the expert system and evaluated from the practical viewpoint. The results show that the neural network system is useful in handling the assortment problem.     

结构经济序列分析的状态空间方法

本文讨论结构经济时间序列用状态空间模型进行分解处理的方法．在§１中综述结构时间序列的状态空间描述．§２中着重论述了将处理不完全数据的ＥＭ－算法应用于状态空间模型参数的极大似然估计．在§３中给出采用本文所述方法对一些我国宏观经济序列的计算实例．     

Stochastic ultimate load analysis:models and solution methods 1

The stochastic ultimate load analysis model used in the safety analysis of engineering structures can be treated as a special case of chance-constrained problems (CCP) which minimize a stochastic cost function subject to some probabilistic constraints. Some special cases (such as a deterministic cost function with probabilistic constraints or deterministic constraints with a random cost function) for ultimate load analysis have airady been investigated by various researchers. In this paper, a generai probabilistic approach to stochastic ultimate load analysis is given. In doing so, some approximation techniques are needed due to the fact that the problems at hand are too complicated to evaluate precisely. We propose two extensions of the SQP method in which the variables appear in the algorithms inexactly. These algorithms are shown to be globally convergent for all models and locally superlinearly convergent for some special cases     

Modeling international investment decisions for financial holding companies

This research analyzes the internationalization process model developed by Johanson and Vahlne and derives two integer programming investment decision models that consider the risk attitudes of investment firms. Johanson and Vahlne’s model provides a starting point for building a model that suits the investment approach and decision making process of financial holding companies. In practice, when firms make an international investment decision, there is a need for a model that can generate outputs based on financial measures such as profit, investment returns, and tolerable levels of risk. Thus, in this paper, Johanson and Vahlne’s concepts are studied and financial managers are interviewed to derive models that match the investment decision procedures of the firms. The model helps firms manage the risks of their investments and derive accurate investment strategies based on investment objectives and constraints.