On the computation of linearly constrained stationary points

We study a numerical method for the computation of linearly constrained stationary points. The proposed method can be interpreted as a projected gradient method with constant stepsize in which one allows perturbations in the admissible set and controls these perturbations in each iteration. The method is applicable to some classes of overdetermined problems to which the projected gradient method may not be directly applicable. Illustrative numerical examples are given.     

On functions whose stationary points are global minima

In this paper, a characterization of functions whose stationary points are global minima is studied. By considering the level sets of a real function as a point-to-set mapping, and by examining its semicontinuity properties, we obtain the result that a real function, defined on a subset ofR ⁿ and satisfying some mild regularity conditions, belongs to the above family iff the point-to-set mapping of its level sets is strictly lower semicontinuous. Mathematical programming applications are also mentioned.The authors are thankful to an anonymous referee of an earlier version of this paper for his valuable comments. This research was partially supported by the Office of Naval Research, Contract No. N-00014-75-C-0267, by the National Science Foundation, Grant No. MPS-71-03341-A03, and by the US Energy Research and Development Administration, Contract No. E(04-3)-326 PA-18. An earlier version of this paper appeared as CORE Discussion Paper No. 7502. Part of this research was carried out while the first author was at CORE, Louvain, Belgium.     

Generalised twists,stationary loops,and the Dirichlet energy over a space of measure preserving maps

Let be a bounded Lipschitz domain and consider the Dirichlet energy functional

On consistency of stationary points of stochastic optimization problems in a Banach space

Recently, Balaji and Xu studied the consistency of stationary points, in the sense of the Clarke generalized gradient, for the sample average approximations to a one-stage stochastic optimization problem in a separable Banach space with separable dual. We present an alternative approach, showing that the restrictive assumptions that the dual space is separable and the Clarke generalized gradient is a (norm) upper semicontinuous and compact-valued multifunction can be dropped. For that purpose, we use two results having independent interest: a strong law of large numbers and a multivalued Komlós theorem in the dual to a separable Banach space, and a result on the weak* closedness of the expectation of a random weak* compact convex set.     

On existence and uniqueness of stationary points in semi-infinite optimization

The paper deals with semi-infinite optimization problems which are defined by finitely many equality constraints and infinitely many inequality constraints. We generalize the concept of strongly stable stationary points which was introduced by Kojima for finite problems; it refers to the local existence and uniqueness of a stationary point for each sufficiently small perturbed problem, where perturbations up to second order are allowed. Under the extended Mangasarian-Fromovitz constraint qualification we present equivalent conditions for the strong stability of a considered stationary point in terms of first and second derivatives of the involved functions. In particular, we discuss the case where the reduction approach is not satisfied. Received June 30, 1995 / Revised version received October 9, 1998? Published online June 11, 1999     

On a new class of nonzero-sum discounted stochastic games having stationary Nash equilibrium points

A new class of nonzero-sum Borel state space discounted stochastic games having stationary Nash equilibria is presented. Some applications to economic theory are also included. Received: January 2002/Revised: July 2002     

On proving existence of feasible points in equality constrained optimization problems

Various algorithms can compute approximate feasible points or approximate solutions to equality and bound constrained optimization problems. In exhaustive search algorithms for global optimizers and other contexts, it is of interest to construct bounds around such approximate feasible points, then to verify (computationally but rigorously) that an actual feasible point exists within these bounds. Hansen and others have proposed techniques for proving the existence of feasible points within given bounds, but practical implementations have not, to our knowledge, previously been described. Various alternatives are possible in such an implementation, and details must be carefully considered. Also, in addition to Hansen’s technique for handling the underdetermined case, it is important to handle the overdetermined case, when the approximate feasible point corresponds to a point with many active bound constraints. The basic ideas, along with experimental results from an actual implementation, are summarized here. This work was supported in part by National Science Foundation grant CCR-9203730.     

On the computation of highly oscillatory multivariate integrals with stationary points

We consider two types of highly oscillatory bivariate integrals with a nondegenerate stationary point. In each case we produce an asymptotic expansion and two kinds of quadrature algorithms: an asymptotic method and a Filon-type method. Our results emphasize the crucial role played by the behaviour at the stationary point and by the geometry of the boundary of the underlying domain. In memory of Germund Dahlquist (1925–2005). AMS subject classification (2000) Primary 65D32     

On stationary points of the implicit Lagrangian for nonlinear complementarity problems

Mangasarian and Solodov have recently introduced an unconstrained optimization problem whose global minima are solutions of the nonlinear complementarity problem (NCP). In this paper, we show that, if the mapping involved in NCP has a positive-definite Jacobian, then any stationary point of the optimization problem actually solves NCP. We also discuss a descent method for solving the unconstrained optimization problem.The authors are indebted to a referee for a helpful suggestion that led them to develop the descent method described in Section 3. They are grateful to Professor F. Facchinei, who kindly pointed out an error in the proof of Theorem 2.3 in an earlier version of the paper. The also thank Professor P. Tseng for a discussion on Theorem 3.1.     

On stationary flows with energy dependent nonlocal viscosities

A nonlocal constitutive law for an incompressible viscous flow in which the viscosity depends on the total dissipation energy of the fluid is obtained as the limit case of very large thermal conductivity when the viscosity varies with the temperature. A rigorous analysis is illustrated within the Hilbertian framework for unidirectional stationary flows of Newtonian and Bingham fluids with heating by viscous dissipation. An extension to quasi-Newtonian fluids of power law type and with temperature dependent viscosities is obtained in the context of the heat equation with an L¹-term. The nonlocal model proposed by Ladyzhenskaya in 1966 as a modification of Navier-Stokes equations can be, in particular, obtained with this procedure. Bibliography: 14 titles.Dedicated to O. A. Ladyzhenskaya on the occasion of her 80th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 99–117.     

On the complexity of finding first-order critical points in constrained nonlinear optimization

The complexity of finding $\epsilon $ -approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that $O(\epsilon ^{-2})$ in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order short-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, and requires minimal assumptions on the objective function. Since a bound of the same order is known to be valid for the unconstrained case, this leads to the conclusion that the presence of possibly nonlinear/nonconvex inequality/equality constraints is irrelevant for this bound to apply.     

The relaxed Dirichlet energy of mappings into a manifold

Let be a smooth 1-connected compact oriented manifold without boundary, such that its 2-homology group has no torsion. We characterize in any dimension n the weak lower semicontinuous envelope of the Dirichlet integral of Sobolev maps in .Received: 6 November 2004, Accepted: 15 November 2004, Published online: 22 December 2004     

Nontrivial stationary points of two-species self-structured communities

The two-species model of self-structured stationary biological communities proposed by U. Dieckmann and R. Law is considered. A way of investigating the system of integro-differential equations describing the model equilibrium is developed, nontrivial stationary points are found, and constraints on the model parameter space resulting in similar stationary points are studied. The results are applied to a number of widely known biological scenarios.     

Implicit functions and sensitivity of stationary points

We consider the spaceL(D) consisting of Lipschitz continuous mappings fromD to the Euclideann-space ⁿ ,D being an open bounded subset of ⁿ . LetF belong toL(D) and suppose that solves the equationF(x) = 0. In case that the generalized Jacobian ofF at is nonsingular (in the sense of Clarke, 1983), we show that forG nearF (with respect to a natural norm) the systemG(x) = 0 has a unique solution, sayx(G), in a neighborhood of Moreover, the mapping which sendsG tox(G) is shown to be Lipschitz continuous. The latter result is connected with the sensitivity of strongly stable stationary points in the sense of Kojima (1980); here, the linear independence constraint qualification is assumed to be satisfied.