No-gap second-order optimality conditions for optimal control problems with a single state constraint and control

The paper deals with optimal control problems with only one control variable and one state constraint, of arbitrary order. We consider the case of finitely many boundary arcs and touch times. We obtain a no-gap theory of second-order conditions, allowing to characterize second-order quadratic growth.     

Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications

We consider a difficult class of optimization problems that we call a mathematical program with vanishing constraints. Problems of this kind arise in various applications including optimal topology design problems of mechanical structures. We show that some standard constraint qualifications like LICQ and MFCQ usually do not hold at a local minimum of our program, whereas the Abadie constraint qualification is sometimes satisfied. We also introduce a suitable modification of the standard Abadie constraint qualification as well as a corresponding optimality condition, and show that this modified constraint qualification holds under fairly mild assumptions. We also discuss the relation between our class of optimization problems with vanishing constraints and a mathematical program with equilibrium constraints.     

The bilevel programming problem: reformulations, constraint qualifications and optimality conditions

We consider the bilevel programming problem and its optimal value and KKT one level reformulations. The two reformulations are studied in a unified manner and compared in terms of optimal solutions, constraint qualifications and optimality conditions. We also show that any bilevel programming problem where the lower level problem is linear with respect to the lower level variable, is partially calm without any restrictive assumption. Finally, we consider the bilevel demand adjustment problem in transportation, and show how KKT type optimality conditions can be obtained under the partial calmness, using the differential calculus of Mordukhovich.     

Geometry of optimality conditions and constraint qualifications

Certain types of necessary optimality conditions for mathematical programming problems are equivalent to corresponding regularity conditions on the constraint set. For any problem, a certain natural optimality condition, dependent upon the particular constraint set, is always satisfied. This condition can be strengthened in numerous ways by invoking appropriate regularity assumptions on the constraint set. Results are presented for Euclidean spaces and some extensions to Banach spaces are given.This work was supported in part by the Office of Naval Research, Contract No. N00014-67-A-0321-0003 (NR-047-095).     

Geometry of optimality conditions and constraint qualifications: The convex case

The cones of directions of constancy are used to derive: new as well as known optimality conditions; weakest constraint qualifications; and regularization techniques, for the convex programming problem. In addition, the badly behaved set of constraints, i.e. the set of constraints which causes problems in the Kuhn—Tucker theory, is isolated and a computational procedure for checking whether a feasible point is regular or not is presented.This research was supported by the National Research Council of Canada and le Gouvernement du Quebec and is part of the author's Ph.D. Dissertation done at McGill University, Montreal, Que., under the guidance of Professor S. Zlobec.     

均衡约束数学规划的约束规格和最优性条件综述

约束规格在约束优化问题的最优性条件中起着重要的作用,介绍了近几年国际上关于均衡约束数学规划(简记为MPEC)的约束规格以及最优性条件的研究成果, 包括以下主要内容: (1) MPEC常用的约束规格(如线性无关约束规格 (MPEC-LICQ)、Mangasarian-Fromovitz约束规格 (MPEC-MFCQ)等)和新的约束规格(如恒秩约束规格、常数正线性相关约束规格等), 以及它们之间的关系; (2) MPEC常用的稳定点; (3) MPEC的最优性条件. 最后还对MPEC的约束规格和最优性条件的研究前景进行了探讨.     

Inventory control using a service constraint on the expected customer order waiting time

In this paper an analytical model is presented which permits the determination of the minimal inventory reorder point subject to (a) a maximum specified expected customer order waiting time or (b) a maximum specified probability of a customer order waiting more than a predetermined time span. The probability distribution of the customer order waiting time is determined for an arbitrary demand distribution (e.g. normal, gamma or exponential) and under stochastic replenishment lead time conditions by use of arguments from renewal theory. The results established can be used for the customer-oriented control of inventory policy as well as for the analysis of multi-echelon inventory systems.     

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

An optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.     

Necessary optimality conditions for nonautonomous control systems with an infinite time horizon

We consider a nonautonomous optimal control problem on an infinite time horizon with an integral functional containing a positive discounting factor. In the case of a dominating discounting factor, we obtain a variant of the Pontryagin maximum principle that contains explicit expressions for the adjoint variable and the Hamiltonian of the problem.     

Locally Lipschitz vector optimization problems: second-order constraint qualifications,regularity condition and KKT necessary optimality conditions

Positivity - In the present paper, we are concerned with a class of constrained vector optimization problems, where the objective functions and active constraint functions are locally Lipschitz at...     

Inventory control based on advanced probability theory,an application

Whenever stock is placed as a buffer between consumption and supply the decision when to replenish the stock is based on uncertain values of future demand and supply variables. Uncertainty exists about the replenishment lead time, about the number of demands and the quantities demanded during this period. We develop a new analytical expression for the reorder point, which is based on the desired service level and three distributions: the distribution of the quantity of single demands during lead time, the distribution of the lengths of time intervals between successive demands, and the distribution of the lead time itself. The distribution of lead time demand is derived from the distributions of individual demand quantities and not from the demand per period. It is not surprising that the resulting formulae for the mean and variance are different from those currently used. The theory developed is also applicable to periodic review systems. The system has been implemented at CERN and enables a significant enhancement of the service level, while reducing the average stock.     

Inventory model of deteriorated items with a constraint: A geometric programming approach

An inventory model for deteriorating items is built-up with limited storage space. Here, demand rate for the items is finite, items deteriorate at constant rates and are replenished instantaneously. Following EOQ model, the problem is formulated with and without truncation on the deterioration term and ultimately is converted to the minimization of a signomial expression with a posynomial constraint. It is solved by modified geometric programming (MGP) method and non-linear programming (NLP) method. The problem is supported by numerical examples. The results from two versions of the model (with and without truncation) and two methods (i.e. MGP and NLP) are compared.     

Control with conic control constraint set

A sufficient condition is given for controllability with conic control constraint set, and a control is computed which performs the transfer to the origin.     

Convergence and quasi-optimality of an adaptive finite element method for optimal control problems with integral control constraint

In this paper we study the convergence of an adaptive finite element method for optimal control problems with integral control constraint. For discretization, we use piecewise constant discretization for the control and continuous piecewise linear discretization for the state and the co-state. The contraction, between two consecutive loops, is proved. Additionally, we find the adaptive finite element method has the optimal convergence rate. In the end, we give some examples to support our theoretical analysis.     

Optimal control problems with unbounded constraint sets

We consider in this paper optimal control problems in which some of the constraint sets are unbounded. Firstly we deal with problems in which the control set is unbounded, so that ‘impulses’ are allowed as admissible controls, discontinuous functions as admissible trajectories. The second type of problem treated is that of infinite horizons, the time set being unbounded. Both class of problems are treated in a similar way. Firstly, a problem is transformed into a semi-infinite linear programming problem by embedding the spacesof admissible trajectory-control pairs into spaces of measures. Then this is mapped into an appropriate nonstandard structure, where a near-minimizer is found for the non-standard optimization; this entity is mapped back as a minimizer for the original problem. An appendix is including introducing the basic concepts of nonstandard analysis

Numerical methods are presented for the estimation of the minimizing measure, and the construction of nearly optimal trajectory-control pairs. Examples are given involving multiplicative controls     

A global existence and uniqueness result for a stochastic Allen–Cahn equation with constraint

This paper addresses the analysis of a time noise‐driven Allen–Cahn equation modelling the evolution of damage in continuum media in the presence of stochastic dynamics. The nonlinear character of the equation is mainly due to a multivoque maximal monotone operator representing a constraint on the damage variable, which is forced to take physically admissible values. By a Yosida approximation and a time‐discretization procedure, we prove a result of global‐in‐time existence and uniqueness of the solution to the stochastic problem. Copyright © 2017 John Wiley & Sons, Ltd.     

Inventory control in the presence of an electronic marketplace

This paper studies a periodic review inventory model in the presence of an electronic marketplace (EM). Emergency orders can be placed in the EM for additional cost, and excess inventory can be sold to the EM. When the order leadtime from the supplier is one period, the optimal inventory control policy is developed from a dynamic programming model of the problem. The policy is characterized by three critical inventory levels. When the order leadtime from the supplier is longer than one period, an EM policy is developed to determine the quantities of inventory to purchase from and sell to the EM in each period. Based on this EM policy, three ordering policies are proposed to determine the order quantity from the supplier. Numerical results show that significant cost reductions can be obtained by using the EM to adjust the inventory level in each period. The amount of cost reduction is greatly affected by system parameters, especially the order leadtime from the supplier and the costs for transactions in the EM.