Stochastic multiobjective problems with complementarity constraints and applications in healthcare management

We consider a class of stochastic multiobjective problems with complementarity constraints (SMOPCCs) in this paper. We derive the first-order optimality conditions including the Clarke/Mordukhovich/strong-type stationarity in the Pareto sense for the SMOPCC. Since these first-order optimality systems involve some unknown index sets, we reformulate them as nonlinear equations with simple constraints. Then, we introduce an asymptotic method to solve these constrained equations. Furthermore, we apply this methodology results to a patient allocation problem in healthcare management.     

Lipschitz Continuity of Approximate Solution Mappings to Parametric Generalized Vector Equilibrium Problems

In this paper, we consider a class of multiobjective problems with equilibrium constraints. Our first task is to extend the existing constraint qualifications for mathematical problems with equilibrium constraints from the single-objective case to the multiobjective case, and our second task is to derive some stationarity conditions under the proper Pareto sense for the considered problem. After doing that, we devote ourselves to investigating the relationships among the extended constraint qualifications and the proper Pareto stationarity conditions.     

A note on stability of stationary points in mathematical programs with generalized complementarity constraints

We consider parameter-dependent mathematical programs with constraints governed by the generalized non-linear complementarity problem and with additional non-equilibrial constraints. We study a local behaviour of stationarity maps that assign the respective C- or M-stationarity points of the problem to the parameter. Using generalized differential calculus rules, we provide criteria for the isolated calmness and the Aubin properties of stationarity maps considered. To this end, we derive and apply formulas of some particular objects of the third-order variational analysis.     

On the Relation Between Two Approaches to Necessary Optimality Conditions in Problems with State Constraints

We consider a class of optimal control problems with a state constraint and investigate a trajectory with a single boundary interval (subarc). Following R.V. Gamkrelidze, we differentiate the state constraint along the boundary subarc, thus reducing the original problem to a problem with mixed control-state constraints, and show that this way allows one to obtain the full system of stationarity conditions in the form of A.Ya. Dubovitskii and A.A. Milyutin, including the sign definiteness of the measure (state constraint multiplier), i.e., the nonnegativity of its density and atoms at junction points. The stationarity conditions are obtained by a two-stage variation approach, proposed in this paper. At the first stage, we consider only those variations, which do not affect the boundary interval, and obtain optimality conditions in the form of Gamkrelidze. At the second stage, the variations are concentrated on the boundary interval, thus making possible to specify the stationarity conditions and obtain the sign of density and atoms of the measure.     

Weak and strong stationarity in generalized bilevel programming and bilevel optimal control

In this article, we consider a general bilevel programming problem in reflexive Banach spaces with a convex lower level problem. In order to derive necessary optimality conditions for the bilevel problem, it is transferred to a mathematical program with complementarity constraints (MPCC). We introduce a notion of weak stationarity and exploit the concept of strong stationarity for MPCCs in reflexive Banach spaces, recently developed by the second author, and we apply these concepts to the reformulated bilevel programming problem. Constraint qualifications are presented, which ensure that local optimal solutions satisfy the weak and strong stationarity conditions. Finally, we discuss a certain bilevel optimal control problem by means of the developed theory. Its weak and strong stationarity conditions of Pontryagin-type and some controllability assumptions ensuring strong stationarity of any local optimal solution are presented.     

Optimal Control of State Constrained Integral Equations

We consider the optimal control problem of a class of integral equations with initial and final state constraints, as well as running state constraints. We prove Pontryagin’s principle, and study the continuity of the optimal control and of the measure associated with first order state constraints. We also establish the Lipschitz continuity of these two functions of time for problems with only first order state constraints.     

On the control of time discretized dynamic contact problems

We consider optimal control problems with distributed control that involve a time-stepping formulation of dynamic one body contact problems as constraints. We link the continuous and the time-stepping formulation by a nonconforming finite element discretization and derive existence of optimal solutions and strong stationarity conditions. We use this information for a steepest descent type optimization scheme based on the resulting adjoint scheme and implement its numerical application.     

Necessary optimality conditions for a special class of bilevel programming problems with unique lower level solution

We consider a bilevel programming problem in Banach spaces whose lower level solution is unique for any choice of the upper level variable. A condition is presented which ensures that the lower level solution mapping is directionally differentiable, and a formula is constructed which can be used to compute this directional derivative. Afterwards, we apply these results in order to obtain first-order necessary optimality conditions for the bilevel programming problem. It is shown that these optimality conditions imply that a certain mathematical program with complementarity constraints in Banach spaces has the optimal solution zero. We state the weak and strong stationarity conditions of this problem as well as corresponding constraint qualifications in order to derive applicable necessary optimality conditions for the original bilevel programming problem. Finally, we use the theory to state new necessary optimality conditions for certain classes of semidefinite bilevel programming problems and present an example in terms of bilevel optimal control.     

Dirichlet Boundary Control of Semilinear Parabolic Equations Part 2: Problems with Pointwise State Constraints

This paper is the continuation of the paper ``Dirichlet boundary control of semilinear parabolic equations. Part 1: Problems with no state constraints.' It is concerned with an optimal control problem with distributed and Dirichlet boundary controls for semilinear parabolic equations, in the presence of pointwise state constraints. We first obtain approximate optimality conditions for problems in which state constraints are penalized on subdomains. Next by using a decomposition theorem for some additive measures (based on the Stone—Cech compactification), we pass to the limit and recover Pontryagin's principles for the original problem. Accepted 21 July 2001. Online publication 21 December 2001.     

Optimality conditions for a class of inverse optimal control problems with partial differential equations

ABSTRACT

We consider bilevel optimization problems which can be interpreted as inverse optimal control problems. The lower-level problem is an optimal control problem with a parametrized objective function. The upper-level problem is used to identify the parameters of the lower-level problem. Our main focus is the derivation of first-order necessary optimality conditions. We prove C-stationarity of local solutions of the inverse optimal control problem and give a counterexample to show that strong stationarity might be violated at a local minimizer.     

Instantaneous control for traffic flow

The solution methods for optimal control problems with coupled partial differential equations as constraints are computationally costly and memory intensive; in particular for problems stated on networks, this prevents the methods from being relevant. We present instantaneous control problems for the optimization of traffic flow problems on road networks. We derive the optimality conditions, investigate the relation to the full optimal control problem and prove that certain properties of the optimal control problem carry over to the instantaneous one. We propose a solution algorithm and compare quality of the computed controls and run‐times. Copyright © 2006 John Wiley & Sons, Ltd.     

Solving quadratic convex bilevel programming problems using a smoothing method

In this paper, we present a smoothing sequential quadratic programming to compute a solution of a quadratic convex bilevel programming problem. We use the Karush-Kuhn-Tucker optimality conditions of the lower level problem to obtain a nonsmooth optimization problem known to be a mathematical program with equilibrium constraints; the complementary conditions of the lower level problem are then appended to the upper level objective function with a classical penalty. These complementarity conditions are not relaxed from the constraints and they are reformulated as a system of smooth equations by mean of semismooth equations using Fisher-Burmeister functional. Then, using a quadratic sequential programming method, we solve a series of smooth, regular problems that progressively approximate the nonsmooth problem. Some preliminary computational results are reported, showing that our approach is efficient.     

The invariant manifolds of systems with first integrals

Several additional possibilities of the Routh–Lyapunov method for isolating and analysing the stationarity sets of dynamical systems admitting of smooth first integrals are discussed. A procedure is proposed for isolating these sets together with the first integrals corresponding to the vector fields for these sets. This procedure is based on solving the stationarity equations of the family of first integrals of the problem in part of the variables and parameters occurring in this family. The effectiveness of this approach is demonstrated for two problems in the dynamics of a rigid body.     

Sensitivity Analysis for Optimal Control Problems Subject to Higher Order State Constraints

A family of parameter dependent optimal control problems is considered. The problems are subject to higher-order inequality type state constraints. It is assumed that, at the reference value of the parameter, the solution exists and is regular. Regularity conditions are formulated under which the original problems are locally equivalent to some other problems subject to equality type constraints only. The classical implicit function theorem is applied to these new problems to investigate Fréchet dif ferentiability of the stationarity points with respect to the parameter.     

Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints

We derive second-order sufficient optimality conditions for discontinuous controls in optimal control problems of ordinary differential equations with initial-final state constraints and mixed state-control constraints of equality and inequality type. Under the assumption that the gradients with respect to the control of active mixed constraints are linearly independent, the sufficient conditions imply a bounded strong minimum in the problem.     

Optimization of a patch

We study optimal patterns of a patch made of an elastic anisotropic homogeneous material for covering a hole in a two-dimensional body possessing different physical characteristics. In addition to the optimization problem for inclusions in two-dimensional and three-dimensional elastic and piezoelectric bodies, we also consider similar problems for an arbitrary formally selfadjoint elliptic system of differential equations in multidimensional domains. A condition for the stationarity of the energy functional is obtained; for a free parameter the matrix of orthogonal transformations of the Euclidean space is taken; the result is based on an algebraic fact about small increments of orthogonal and unitary matrices. Bibliography: 23 titles. Illustrations: 1 figure.     

On the convergence of inexact augmented Lagrangian methods for problems with convex constraints

We consider the augmented Lagrangian method (ALM) for constrained optimization problems in the presence of convex inequality and convex abstract constraints. We focus on the case where the Lagrangian sub-problems are solved up to approximate stationary points, with increasing accuracy. We analyze two different criteria of approximate stationarity for the sub-problems and we prove the global convergence to stationary points of ALM in both cases.     

Approximation of optimal control problems with state constraints

We study the approximation of control problems governed by elliptic partial differential equations with pointwise state constraints. For a finite dimensional approximation of the control set and for suitable perturbations of the state constraints, we prove that the corresponding sequence of discrete control problems converges to a relaxed problem. A similar analysis is carried out for problems in which the state equation is discretized by a finite element method.     

Optimal Control of Differential–Algebraic Equations of Higher Index, Part 1: First-Order Approximations

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of these problems which can be transformed to index one control problems. For this class of higher index DAEs, we derive first-order approximations and adjoint equations for the functionals defining the problem. These adjoint equations are then used to state, in the accompanying paper, the necessary optimality conditions in the form of a weak maximum principle. The constructive way used to prove these optimality conditions leads to globally convergent algorithms for control problems with state constraints and defined by higher index DAEs.     

Verification of Second-Order Sufficient Optimality Conditions for Semilinear Elliptic and Parabolic Control Problems

We study optimal control problems for semilinear parabolic equations subject to control constraints and for semilinear elliptic equations subject to control and state constraints. We quote known second-order sufficient optimality conditions (SSC) from the literature. Both problem classes, the parabolic one with boundary control and the elliptic one with boundary or distributed control, are discretized by a finite difference method. The discrete SSC are stated and numerically verified in all cases providing an indication of optimality where only necessary conditions had been studied before.