Optimal control problems with state constraints for semilinear distributed-parameter systems

We consider optimal control problems for distributed-parameter systems described by semilinear equations, with constraints on the control and on the state, and an exact pointwise target condition. As an application of a general theory of nonlinear programming problems in Banach spaces, a version of the Pontryagin maximum principle is obtained.This research was partly supported by the National Science Foundation under Grant DMS-92-21819.     

A Scalable Parallel Interior Point Algorithm for Stochastic Linear Programming and Robust Optimization

We present a computationally efficient implementation of an interior point algorithm for solving large-scale problems arising in stochastic linear programming and robust optimization. A matrix factorization procedure is employed that exploits the structure of the constraint matrix, and it is implemented on parallel computers. The implementation is perfectly scalable. Extensive computational results are reported for a library of standard test problems from stochastic linear programming, and also for robust optimization formulations.The results show that the codes are efficient and stable for problems with thousands of scenarios. Test problems with 130 thousand scenarios, and a deterministic equivalent linear programming formulation with 2.6 million constraints and 18.2 million variables, are solved successfully.     

A General Method of Solution for Special Structure Linear Programmes

The calculation routines of simplex, transportation, aircraft routing, generalized, capacitated and multi-index problems are shown to be special cases of a general method of linear programming. The relationship between these routines is shown and the formulation enables new methods to be created.     

线性规划的最钝角松弛算法

本文提出一个基于最钝角原理的松弛算法求解线性规划问题。该算法依据最钝角原理略去部分约束得到一个规模较小的子问题,用原始单纯形算法解之;再添加所略去的约束恢复原问题,若此时全部约束条件均满足则已获得一个基本最优解,否则用对偶单纯形算法继续求解。初步的数值试验表明,新算法比传统两阶段单纯形算法快得多。     

Normality of the maximum principle for nonconvex constrained Bolza problems

We consider a Bolza optimal control problem with state constraints. It is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle. In general, the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls. In the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate, cf. [A.V. Arutyanov, S.M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim. 35 (1997) 930–952; F. Rampazzo, R.B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA 16 (4) (1999) 335–351; F. Rampazzo, R.B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim. 39 (4) (2000) 989–1007]. Our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle. In particular we allow the initial condition to be fixed and the state constraints to be nonsmooth. To prove normality we use J. Yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined, in turn, by linearization of the original set of constraints along an extremal trajectory.     

解可分离约束双凹规划问题的一种外逼近方法[英文]

在本文中，我们提出了双凹规划问题和更一般的广义凹规划问题。我们给出了双凹规划问题的整体最优性条件，并构造了一个有限终止外逼近算法。     

Forward differential dynamic programming

The dynamic programming formulation of the forward principle of optimality in the solution of optimal control problems results in a partial differential equation with initial boundary condition whose solution is independent of terminal cost and terminal constraints. Based on this property, two computational algorithms are described. The first-order algorithm with minimum computer storage requirements uses only integration of a system of differential equations with specified initial conditions and numerical minimization in finite-dimensional space. The second-order algorithm is based on the differential dynamic programming approach. Either of the two algorithms may be used for problems with nondifferentiable terminal cost or terminal constraints, and the solution of problems with complicated terminal conditions (e.g., with free terminal time) is greatly simplified.     

Duality and Correction of Inconsistent Constraints for Improper Linear Programming Problems

We continue the study of approximation properties of alternative duality schemes for improper problems of linear programming. The schemes are based on the use of the classical Lagrange function regularized simultaneously in primal and dual variables. The earlier results on the connection of its saddle points with the lexicographic correction of the right-hand sides of constraints in improper problems of the first and second kind are transferred to a more general type of improperness. Convergence theorems are presented and an informal interpretation of the obtained generalized solution is given.     

Equilibrium constrained optimization problems

We consider equilibrium constrained optimization problems, which have a general formulation that encompasses well-known models such as mathematical programs with equilibrium constraints, bilevel programs, and generalized semi-infinite programming problems. Based on the celebrated KKM lemma, we prove the existence of feasible points for the equilibrium constraints. Moreover, we analyze the topological and analytical structure of the feasible set. Alternative formulations of an equilibrium constrained optimization problem (ECOP) that are suitable for numerical purposes are also given. As an important first step for developing efficient algorithms, we provide a genericity analysis for the feasible set of a particular ECOP, for which all the functions are assumed to be linear.     

Performance-Driven Layer Assignment by Integer Linear Programming and Path-Constrained Hypergraph Partitioning

Performance-driven physical layout design is becoming increasingly important for both high speed integrated circuits and printed circuit boards. This paper studies the problem of assigning wire segments into two layers so as to minimize the number of vias, while taking into account performance constraints such as layer preference and circuit timing. We show that using the Elmore delay model, three timing problems in synchronous digital circuits—the long path problem, the short path problem and the time skew problem—can be formulated as a set of linear inequalities. We use the model of signed hypergraph to represent two-layer routings and formulate the performance-driven optimum layer assignment problem as the path-constrained maximum balance problem in a signed hypergraph. Two solution methods are developed and implemented. First, an integer linear programming formulation is derived for finding exact solutions. Second, a local-search heuristic for hypergraph partitioning is extended to cope with path-inequality constraints. Experimental results on a set of layer-assignment benchmarks demonstrated that the path-constrained local-search heuristic achieves optimum or near-optimum solutions with several orders of magnitude faster than the integer linear programming approach.     

A Unified Monotonic Approach to Generalized Linear Fractional Programming

We present an efficient unified method for solving a wide class of generalized linear fractional programming problems. This class includes such problems as: optimizing (minimizing or maximizing) a pointwise maximum or pointwise minimum of a finite number of ratios of linear functions, optimizing a sum or product of such ratios, etc. – over a polytope. Our approach is based on the recently developed theory of monotonic optimization.     

局部FC-一致空间内具有联立广义矢量拟平衡组约束的数学规划(英文)

在局部FC-一致空间内引入和研究了某些新的联立广义矢量拟平衡问题组和具有联立广义矢量拟平衡组约束的数学规划.应用作者在局部FC-一致空间得到的Himmelberg型不动点定理,首先在局部FC-一致空间内对联立广义矢量拟平衡问题组的解证明了某些新的存在性定理.作为应用,对具有联立广义矢量拟平衡组约束的数学规划的解得到了某些新的存在性定理.     

Efficiency Conditions and Duality for a Class of Multiobjective Fractional Programming Problems

A class of constrained multiobjective fractional programming problems is considered from a viewpoint of the generalized convexity. Some basic concepts about the generalized convexity of functions, including a unified formulation of generalized convexity, are presented. Based upon the concept of the generalized convexity, efficiency conditions and duality for a class of multiobjective fractional programming problems are obtained. For three types of duals of the multiobjective fractional programming problem, the corresponding duality theorems are also established.     

Linear Formulation of a Distributed Boundary Control Problem

In this paper, we consider a distributed boundary control problem governed by an elliptic partial differential equation with state constraints and a minimax objective function. The continuous optimal control problem, discretized with the finite element method, is numerically approximated by a family of linear programming problems. Application to an optimal configuration problem is discussed.     

Stochastic programming problems with generalized integrated chance constraints

If the constraints in an optimization problem are dependent on a random parameter, we would like to ensure that they are fulfilled with a high level of reliability. The most natural way is to employ chance constraints. However, the resulting problem is very hard to solve. We propose an alternative formulation of stochastic programs using penalty functions. The expectations of penalties can be left as constraints leading to generalized integrated chance constraints, or incorporated into the objective as a penalty term. We show that the penalty problems are asymptotically equivalent under quite mild conditions. We discuss applications of sample-approximation techniques to the problems with generalized integrated chance constraints and propose rates of convergence for the set of feasible solutions. We will direct our attention to the case when the set of feasible solutions is finite, which can appear in integer programming. The results are then extended to the bounded sets with continuous variables. Additional binary variables are necessary to solve sample-approximated chance-constrained problems, leading to a large mixed-integer non-linear program. On the other hand, the problems with penalties can be solved without adding binary variables; just continuous variables are necessary to model the penalties. The introduced approaches are applied to the blending problem leading to comparably reliable solutions.     

A maximum principle for nonsmooth optimal-control problems with state constraints

Optimality conditions are derived in the form of a maximum principle governing solutions to an optimal control problem which involves state constraints. The conditions, which apply in the absence of differentiability assumptions on the data, are stated in terms of Clarke's generalized Jacobians. Although not the most general available, the conditions are derived by a novel method: this involves removal of the state constraints by introduction of a penalty term and application of Ekeland's variational principle.     

Maximum principle in the problem of time optimal response with nonsmooth constraints : PMM vol. 40, n 6, 1976, pp. 1014–1023

The problem of optimal response [1, 2] with nonsmooth (generally speaking, nonfunctional) constraints imposed on the state variables is considered. This problem is used to illustrate the method of proving the necessary conditions of optimality in the problems of optimal control with phase constraints, based on constructive approximation of the initial problem with constraints by a sequence of problems of optimal control with constraint-free state variables. The variational analysis of the approximating problems is carried out by means of a purely algebraic method involving the formulas for the incremental growth of a functional [3, 4] and the theorems of separability of convex sets is not used.Using a passage to the limit, the convergence of the approximating problems to the initial problem with constraints is proved, and for general assumptions the necessary conditions of optimality resembling the Pontriagin maximum principle [1] are derived for the generalized solutions of the initial problem. The conditions of transversality are expressed, in the case of nonsmooth (nonfunctional) constraints by a novel concept of a cone conjugate to an arbitrary closed set of a finite-dimensional space. The concept generalizes the usual notions of the normal and the normal cone for the cases of smooth and convex manifolds.     

A class of generalized variable penalty methods for nonlinear programming

A class of generalized variable penalty formulations for solving nonlinear programming problems is presented. The method poses a sequence of unconstrained optimization problems with mechanisms to control the quality of the approximation for the Hessian matrix, which is expressed in terms of the constraint functions and their first derivatives. The unconstrained problems are solved using a modified Newton's algorithm. The method is particularly applicable to solution techniques where an approximate analysis step has to be used (e.g., constraint approximations, etc.), which often results in the violation of the constraints. The generalized penalty formulation contains two floating parameters, which are used to meet the penalty requirements and to control the errors in the approximation of the Hessian matrix. A third parameter is used to vary the class of standard barrier or quasibarrier functions, forming a branch of the variable penalty formulation. Several possibilities for choosing such floating parameters are discussed. The numerical effectiveness of this algorithm is demonstrated on a relatively large set of test examples.The author is thankful for the constructive suggestions of the referees.     

Linear Programming on Cray Supercomputers

In this note we present computational experience with a linear programming code on two Cray super-computers (Cray-1S, Cray X-MP/48). Results are given for problems involving up to 640,000 variables (1600 constraints) and 10,000 constraints (99,969 variables).     

Maximum Principle for Stochastic Control in Continuous Time with Hard End Constraints

A maximum principle is proved for certain problems of continuous–time stochastic control with hard end constraints (end constraints satisfied a.s.). In the problems, the dynamics (the state differential equation) changes at certain stochastic points in time.