An augmented Lagrangian technique for variational inequalities

A general framework for the treatment of a class of elliptic variational inequalities by an augmented Lagrangian method, when inequalities with infinite-dimensional image space are augmented, is developed. Applications to the obstacle problem, the elastoplastic torsion problem, and the Signorini problem are given.The research of the first author was supported in part by the Air Force Office of Scientific Research under Grants AFOSR-84-0398 and AFOSR-85-0303, by the National Aeronautics and Space Administration under Grant NAG-1-1517, and by NSF under Grant No. UINT-8521208. The second author's research was supported in part by the Fonds zur Förderung der wissenschaftlichen Forschung under S3206 and P6005.     

A generalization of a minimax theorem of Fan via a theorem of the alternative

A minimization problem for a functional on a convex subsetC of a normed linear space is considered. Under certain hypotheses, optimality in a certain subset ofC implies the validity of first-order necessary optimality conditions for the problem inC. The result is applied to a problem in optimal periodic control of neutral functional differential equations.This work was partially supported by a grant from Deutsche Forschungsgemeinschaft and by AFOSR under Grant No. AFOSR-84-0398.     

On the convergence of kernel estimators of probability density functions

Summary The properties of the characteristic function of the fixed-bandwidth kernel estimator of a probability density function are used to derive estimates of the rate of almost sure convergence of such estimators in a family of Hilbert spaces. The convergence of these estimators in a reproducing-kernel Hilbert space is used to prove the uniform convergence of variable-bandwidth estimators. An algorithm employing the fast Fourier transform and heuristic estimates of the optimal bandwidth is presented, and numerical experiments using four density functions are described. This research was supported by the United States Air Force, Air Force Office of Scientific Research, Under Grant Number AFOSR-76-2711.     

Optimality,stability, and convergence in nonlinear control

Sufficient optimality conditions for infinite-dimensional optimization problems are derived in a setting that is applicable to optimal control with endpoint constraints and with equality and inequality constraints on the controls. These conditions involve controllability of the system dynamics, independence of the gradients of active control constraints, and a relatively weak coercivity assumption for the integral cost functional. Under these hypotheses, we show that the solution to an optimal control problem is Lipschitz stable relative to problem perturbations. As an application of this stability result, we establish convergence results for the sequential quadratic programming algorithm and for penalty and multiplier approximations applied to optimal control problems.This research was supported by the U.S. Army Research Office under Contract. Number DAAL03-89-G-0082, by the National Science Foundation under Grant Number DMS 9404431, and by Air Force Office of Scientific Research under Grant Number AFOSR-88-0059. A. L. Dontchev is on leave from the Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria.     

Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems

We develop an abstract framework and convergence theory for Galerkin approximation for inverse problems involving the identification of nonautonomous, in general nonlinear, distributed parameter systems. We provide a set of relatively easily verified conditions which are sufficient to guarantee the existence of optimal solutions and their approximation by a sequence of solutions to a sequence of approximating finite-dimensional identification problems. Our approach is based upon the theory of monotone operators in Banach spaces and is applicable to a reasonably broad class of nonlinear distributed systems. Operator theoretic and variational techniques are used to establish a fundamental convergence result. An example involving evolution systems with dynamics described by nonstationary quasi-linear elliptic operators along with some applications and numerical results are presented and discussed.Part of this research was carried out while the first and third authors were visiting scientists at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center. Hampton, VA, which is operated under NASA Contracts NAS1-17070 and NAS1-18107. Also, a portion of this research was carried out with computational resources made available through a grant to the second and third authors from the San Diego Supercomputer Center operated for the National Science Foundation by General Atomics, San Diego, CA. The research of the first author was supported in part under Grants NSF MCS-8504316, NASA NAG-1-517, AFOSR-84-0398, and AFOSR-F49620-86-C-0111. The second author's research was supported in part by the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund. The research of the third author was supported in part under Grants AFOSR-84-0393 and AFOSR-87-0356.     

Geometric convergence of rational approximations toe −z in infinite sectors

Summary In this paper, we study the geometric convergence of rational approximations toe ^–z in infinite sectors symmetric about the positive real axis.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Florida Research CouncilResearch supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Atomic Energy Commission under Grant AT (11-1)-2075     

Approximation of infinite delay and Volterra type equations

Summary We consider a class of infinite delay equations of neutral type which includes Volterrra type integral und integrodifferential equations. Using abstract approximation results (cf. Trotter-Kato-type) for strongly continuous semigroups we develop an approximation scheme which is based on approximation of the system state by Laguerre (and Legendre) polynomials. Numerical examples demonstrate the feasibility of the scheme and show infinite order convergence for smooth data.Supported in part by the Air Force Office of Scientific Research under Contracts AFORS-84-0398 and AFORS-85-0303 and the National Aeronautics and Space Administration under NASA Grant NAG-1-517 and by NSF under Grant UINT-8521208Supported in part by the Air Force Office of Scientific Research under Contract AFORS-84-0398 and in part by the Fonds zur Förderung der wissenschaftlichen Forschung, Austria, under project No. S3206     

Strong stability of quasi-affine transforms of contraction semigroups and the steady-state riccati equation

This paper studies strong stability of the class of strongly continuous Hilbert space semigroups which are quasi-affine transforms of contraction semigroups. Sufficient conditions for such a semigroup to be approximately strong stable are given. Applications to the stabilizability problem of Hilbert space semigroups, using a feedback involving a solution of the steady-state Riccati equation, are made. Our key tool is the generalization of the LaSalle invariance principle by Hale.This research was supported by the United States Air Force Office of Scientific Research, Directorate of Mathematical and Information Sciences, under Grant No. AFOSR-79-0053D.     

A uniformly differentiable approximation scheme for delay systems using splines

A new spline-based scheme is developed for linear retarded functional differential equations within the framework of semigroups on the Hilbert spaceR ⁿ ×L ². The approximating semigroups inherit in a uniform way the characterization for differentiable semigroups from the solution semigroup of the delay system (e.g., among other things the logarithmic sectorial property for the spectrum). We prove convergence of the scheme in the state spacesR ⁿ ×L ² andH ¹. The uniform differentiability of the approximating semigroups enables us to establish error estimates including quadratic convergence for certain classes of initial data. We also apply the scheme for computing the feedback solutions to linear quadratic optimal control problems.Work done by K. Ito was supported by AFOSR under Contract No. F-49620-86-C-0111, by NASA under Grant No. NAG-1-517, and by NSF under Grant No. UINT-8521208. Work done by F. Kappel was supported by AFOSR under Grant No. 84-0398 and by FWF(Austria) under Grants S3206 and P6005.     

On conditions to have bounded multipliers in locally lipschitz programming

For a nonlinear programming problem with locally Lipschitz objective and inequality constraint functions and continuously differentiable equality constraint functions, a necessary and sufficient condition is presented for the set of multiplier vectors to be nonempty and bounded.This author's work was supported by the National Science Foundation under Grant No. MCS 78-06716.     

Adaptive control of Markov processes with incomplete state information and unknown parameters

Recent results for parameter-adaptive Markov decision processes (MDP's) are extended to partially observed MDP's depending on unknown parameters. These results include approximations converging uniformly to the optimal reward function and asymptotically optimal adaptive policies.This research was supported in part by the Consejo del Sistema Nacional de Educación Tecnologica (COSNET) under Grant 178/84, in part by the Air Force Office of Scientific Research under Grant AFOSR-84-0089, in part by the National Science Foundation under Grant ECS-84-12100, and in part by the Joint Services Electronics Program under Contract F49602-82-C-0033.     

Optimal filter design subject to output sidelobe constraints: Theoretical considerations

The design of filters for detection and estimation in radar and communications systems is considered, with inequality constraints on the maximum output sidelobe levels. A constrained optimization problem in Hilbert space is formulated, incorporating the sidelobe constraints via a partial ordering of continuous functions. Generalized versions (in Hilbert space) of the Kuhn-Tucker and duality theorems allow the reduction of this problem to an unconstrained one in the dual space of regular Borel measures.This research was supported by the National Science Foundation under Grant No. GK-2645, by the National Aeronautics and Space Administration under Grant No. NGL-22-009(124), and by the Australian Research Grants Committee. The authors wish to express their gratitude to Dr. Robert McAulay and Professor Ian Rhodes for various comments and suggestions.     

Informational uniqueness of closed-loop Nash equilibria for a class of nonstandard dynamic games

This paper discusses an extension of the currently available theory of noncooperative dynamic games to game models whose state equations are of order higher than one. In a discrete-time framework, it first elucidates the reasons why the theory developed for first-order systems is not applicable to higher-order systems, and then presents a general procedure to obtain an informationally unique Nash equilibrium solution in the presence of random disturbances. A numerical example solved in the paper illustrates the general approach.Dedicated to G. LeitmannResearch that led to this paper was supported in part by the Office of Naval Research under Contract No N00014-82-K-0469 and in part by the U.S. Air Force under Grant No. AFOSR-84-0054.     

Sequential stable Kuhn-Tucker theorem in nonlinear programming

A general parametric nonlinear mathematical programming problem with an operator equality constraint and a finite number of functional inequality constraints is considered in a Hilbert space. Elements of a minimizing sequence for this problem are formally constructed from elements of minimizing sequences for its augmented Lagrangian with values of dual variables chosen by applying the Tikhonov stabilization method in the course of solving the corresponding modified dual problem. A sequential Kuhn-Tucker theorem in nondifferential form is proved in terms of minimizing sequences and augmented Lagrangians. The theorem is stable with respect to errors in the initial data and provides a necessary and sufficient condition on the elements of a minimizing sequence. It is shown that the structure of the augmented Lagrangian is a direct consequence of the generalized differentiability properties of the value function in the problem. The proof is based on a “nonlinear” version of the dual regularization method, which is substantiated in this paper. An example is given illustrating that the formal construction of a minimizing sequence is unstable without regularizing the solution of the modified dual problem.     

A transformation technique for optimal control problems with partially linear state inequality constraints

This paper considers optimal control problems involving the minimization of a functional subject to differential constraints, terminal constraints, and a state inequality constraint. The state inequality constraint is of a special type, namely, it is linear in some or all of the components of the state vector.A transformation technique is introduced, by means of which the inequality-constrained problem is converted into an equality-constrained problem involving differential constraints, terminal constraints, and a control equality constraint. The transformation technique takes advantage of the partial linearity of the state inequality constraint so as to yield a transformed problem characterized by a new state vector of minimal size. This concept is important computationally, in that the computer time per iteration increases with the square of the dimension of the state vector.In order to illustrate the advantages of the new transformation technique, several numerical examples are solved by means of the sequential gradient-restoration algorithm for optimal control problems involving nondifferential constraints. The examples show the substantial savings in computer time for convergence, which are associated with the new transformation technique.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-76-3075, and by the National Science Foundation, Grant No. MCS-76-21657.     

Optimization algorithm with probabilistic estimation

In this paper, we present a stochastic optimization algorithm based on the idea of the gradient method which incorporates a new adaptive-precision technique. Because of this new technique, unlike recent methods, the proposed algorithm adaptively selects the precision without any need for prior knowledge on the speed of convergence of the generated sequence. With this new technique, the algorithm can avoid increasing the estimation precision unnecessarily, yet it retains its favorable convergence properties. In fact, it tries to maintain a nice balance between the requirements for computational accuracy and those for computational expediency. Furthermore, we present two types of convergence results delineating under what assumptions what kinds of convergence can be obtained for the proposed algorithm.The work reported here was supported in part by NSF Grant No. ECS-85-06249 and USAF Grant No. AFOSR-89-0518. The authors wish to thank the anonymous reviewers whose careful reading and criticism have helped them improve the paper considerably.     

On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces

In a Hilbert space, we study the finite termination of iterative methods for solving a monotone variational inequality under a weak sharpness assumption. Most results to date require that the sequence generated by the method converges strongly to a solution. In this paper, we show that the proximal point algorithm for solving the variational inequality terminates at a solution in a finite number of iterations if the solution set is weakly sharp. Consequently, we derive finite convergence results for the gradient projection and extragradient methods. Our results show that the assumption of strong convergence of sequences can be removed in the Hilbert space case.     

Interior-point ℓ 2-penalty methods for nonlinear programming with strong global convergence properties

We propose two line search primal-dual interior-point methods for nonlinear programming that approximately solve a sequence of equality constrained barrier subproblems. To solve each subproblem, our methods apply a modified Newton method and use an ℓ₂-exact penalty function to attain feasibility. Our methods have strong global convergence properties under standard assumptions. Specifically, if the penalty parameter remains bounded, any limit point of the iterate sequence is either a Karush-Kuhn-Tucker (KKT) point of the barrier subproblem, or a Fritz-John (FJ) point of the original problem that fails to satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ); if the penalty parameter tends to infinity, there is a limit point that is either an infeasible FJ point of the inequality constrained feasibility problem (an infeasible stationary point of the infeasibility measure if slack variables are added) or a FJ point of the original problem at which the MFCQ fails to hold. Numerical results are given that illustrate these outcomes. Research supported by the Presidential Fellowship of Columbia University. Research supported in part by NSF Grant DMS 01-04282, DOE Grant DE-FG02-92EQ25126 and DNR Grant N00014-03-0514.     

Competing symmetries,the logarithmic HLS inequality and Onofri's inequality ons n

The sharp version of the logarithmic Hardy-Littlewood-Sobolev inequality including the cases of equality is established. We then show that this implies Beckner's generalization of Onofri's inequality to arbitrary dimensions and determines the cases of equality.The work of the second author is partially supported by the U.S. National Science Foundation under Grant DMS-90-05729.     

An analysis of reduced Hessian methods for constrained optimization

We study the convergence properties of reduced Hessian successive quadratic programming for equality constrained optimization. The method uses a backtracking line search, and updates an approximation to the reduced Hessian of the Lagrangian by means of the BFGS formula. Two merit functions are considered for the line search: the ₁ function and the Fletcher exact penalty function. We give conditions under which local and superlinear convergence is obtained, and also prove a global convergence result. The analysis allows the initial reduced Hessian approximation to be any positive definite matrix, and does not assume that the iterates converge, or that the matrices are bounded. The effects of a second order correction step, a watchdog procedure and of the choice of null space basis are considered. This work can be seen as an extension to reduced Hessian methods of the well known results of Powell (1976) for unconstrained optimization.This author was supported, in part, by National Science Foundation grant CCR-8702403, Air Force Office of Scientific Research grant AFOSR-85-0251, and Army Research Office contract DAAL03-88-K-0086.This author was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contracts W-31-109-Eng-38 and DE FG02-87ER25047, and by National Science Foundation Grant No. DCR-86-02071.