Existence, uniqueness and qualitative properties of minima to radially symmetric non-coercive non-convex variational problems

We are concerned with the problem of existence, uniqueness and qualitative properties of solutions to the radially symmetric variational problem where is the ball of centered at the origin and with radius , the map is a normal integrand, and is a convex function of the second variable. This kind of problems, with non-convex lagrangians with respect to , arise in various fields of applied sciences, such as optimal design and nonlinear elasticity. Received June 18, 1998; in final form August 26, 1999 / Published online September 14, 2000     

Regularization of convex variational problems with applications to generalized Newtonian fluids

We study variational problems with convex integrands of very general structure by introducing certain regularizations leading to particular minimizers. In a second part we apply the method to stationary generalized Newtonian fluids which gives the existence of solutions under weak hypotheses on the dissipative potential. In particular the potential is not required to be a strictly convex function.Received: 29 July 2003     

Partially finite convex programming,Part I: Quasi relative interiors and duality theory

We study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. We use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable.     

Generalized convex relations with applications to optimization and models of economic dynamics

We examine a notion of generalized convex set-valued mapping, extending the notions of a convex relation and a convex process. Under general conditions, we establish duality results for composite set-valued mappings and for convex programming problems involving convex set-valued mappings. We also present applications to the study of economic dynamical systems, by obtaining the characteristics of optimal paths generated by convex processes, and to optimization problems of a certain class of positively homogeneous increasing functions.     

Regularity properties of optimal controls for problems with time-varying state and control constraints

In this paper we report new results on the regularity of optimal controls for dynamic optimization problems with functional inequality state constraints, a convex time-dependent control constraint and a coercive cost function. Recently, it has been shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters, provided the control constraint set is independent of the time variable. We show that, if the control constraint set, regarded as a time-dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstances, however, a weaker Hölder continuity-like regularity property can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time-varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities.     

A posteriori error estimates for mixed finite element solutions of convex optimal control problems

In this paper, we present an a posteriori error analysis for mixed finite element approximation of convex optimal control problems. We derive a posteriori error estimates for the coupled state and control approximations under some assumptions which hold in many applications. Such estimates can be used to construct reliable adaptive mixed finite elements for the control problems.     

Generic power convergence of operators in banach spaces

Let K be a closed convex subset of a Banach space X. We consider complete metric spaces of self-mappings of K which are nonexpansive with respect to a convex function on X. We prove that the iterates of a generic operator in these spaces converge strongly. In some cases the limits do not depend on the initial points and are the unique fixed point of the operator.     

A posteriori error estimates for hp finite element solutions of convex optimal control problems

In this paper, we present a posteriori error analysis for hp finite element approximation of convex optimal control problems. We derive a new quasi-interpolation operator of Clément type and a new quasi-interpolation operator of Scott-Zhang type that preserves homogeneous boundary condition. The Scott-Zhang type quasi-interpolation is suitable for an application in bounding the errors in L²-norm. Then hp a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems.     

Comparing two methods of resolving homogeneous differential inclusions

Given a compact set we consider the differential inclusion We show how to use the main idea of the method of convex integration [ N], [G], [K] (to control convergence of the gradients of a sequence of approximate solutions by appropriate selection of the sequence) to obtain an optimal existence result. We compare this result with the ones available by the Baire category approach applied to the set of admissible functions with topology. A byproduct of our result is attainment in the minimization problems with integrands L having quasiaffine quasiconvexification that was, in fact, the reason of our interest to differential inclusions. This result can be considered as a first step towards characterization of those minimization problems which are solvable for all boundary data. This problem was solved in [S1] in the scalar case m=1. Received November 5, 1998 / Accepted July 17, 2000 / Published online December 8, 2000     

On the minimum problem for nonconvex, multiple integrals of product type

We consider the problem of minimizing multiple integrals of product type, i.e. where is a bounded, open set in , is a possibly nonconvex, lower semicontinuous function with p-growth at infinity for some and the boundary datum is in (or simply in if ). Assuming that the convex envelope off is affine on each connected component of the set , we prove attainment for () for every continuous, positively bounded below function g such that (i) every point is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that the class of coefficents g that yield existence to () is dense in the space of continuous, positive functions on . We present examples which show that these conditions for attainment are essentially sharp. Received April 12, 2000 / Accepted May 9, 2000 / Published online November 9, 2000     

Optimality conditions for non-finite valued convex composite functions

Burke (1987) has recently developed second-order necessary and sufficient conditions for convex composite optimization in the case where the convex function is finite valued. In this note we present a technique for reducing the infinite valued case to the finite valued one. We then use this technique to extend the results in Burke (1987) to the case in which the convex function may take infinite values. We conclude by comparing these results with those established by Rockafellar (1989) for the piecewise linear-quadratic case.Dedicated to the memory of Robin W. ChaneyResearch supported in part by the National Science Foundation under grants DMS-8602399 and DMS-8803206, and by the Air Force Office of Scientific Research under grant ISSA-860080.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983.     

Bauer's maximum principle and hulls of sets

We use the Bauer maximum principle for quasiconvex, polyconvex and rank-one convex functions to derive Krein-Milman-type theorems for compact sets in . Further we show that in general the set of quasiconvex extreme points is not invariant under transposition and it is different from the set of rank-one convex extreme points. Finally, a set in with different polyconvex, quasiconvex and rank-one convex hulls is constructed. Received September 14, 1999 / Accepted January 14, 2000 /Published online July 20, 2000     

The Generalized Moment Problem with Complexity Constraint

In this paper, we present a synthesis of our differentiable approach to the generalized moment problem, an approach which begins with a reformulation in terms of differential forms and which ultimately ends up with a canonically derived, strictly convex optimization problem. Engineering applications typically demand a solution that is the ratio of functions in certain finite dimensional vector space of functions, usually the same vector space that is prescribed in the generalized moment problem. Solutions of this type are hinted at in the classical text by Krein and Nudelman and stated in the vast generalization of interpolation problems by Sarason. In this paper, formulated as generalized moment problems with complexity constraint, we give a complete parameterization of such solutions, in harmony with the above mentioned results and the engineering applications. While our previously announced results required some differentiability hypotheses, this paper uses a weak form involving integrability and measurability hypotheses that are more in the spirit of the classical treatment of the generalized moment problem. Because of this generality, we can extend the existence and well-posedness of solutions to this problem to nonnegative, rather than positive, initial data in the complexity constraint. This has nontrivial implications in the engineering applications of this theory. We also extend this more general result to the case where the numerator can be an arbitrary positive absolutely integrable function that determines a unique denominator in this finite-dimensional vector space. Finally, we conclude with four examples illustrating our results.     

Duality for Anticonvex Programs

Calling anticonvex a program which is either a maximization of a convex function on a convex set or a minimization of a convex function on the set of points outside a convex subset, we introduce several dual problems related to each of these problems. We give conditions ensuring there is no duality gap. We show how solutions to the dual problems can serve to locate solutions of the primal problem.     

A-B quasiconvexity and implicit partial differential equations

The study of existence of solutions of boundary-value problems for differential inclusions where , is an open subset of , is a compact set, and B is a -valued first order differential operator, is undertaken. As an application, minima of the energy for large magnetic bodies where the magnetization is taken with values on the unit sphere is the induced magnetic field satisfying and is the anisotropic energy density, and the applied external magnetic field is given by , are fully characterized. Setting with , it is shown that E admits a minimizer with if and only if either 0 is on a face of or , where denotes the convex hull of Z. Received: 6 November 2000 / Accepted: 23 January 2001 / Published online: 23 April 2001     

Characterization of optimal shapes and masses through Monge-Kantorovich equation

We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented. Received February 10, 2000 / final version received July 21, 2000?Published online November 8, 2000     

An open mapping theorem using lower semicontinuous functions

We give a simple and direct proof, using lower semicontinuous functions, of a generalization of the open mapping theorem for metrizable topological vector spaces (that are not necessarily locally convex) and operators with complete graph. Our result is in a form more applicable to applied (convex) analysis.     

A regularity theory for variational integrals with L\ln L-Growth

In the present paper we study regularity for local minimizers of the convex variational integral defined on certain classes of vector–valued functions . The underlying energy spaces are natural from the point of view of existence theory. We then show that local minimizers are of class apart from a closed singular set with vanishing Lebesgue measure, provided . For twodimensional problems we obtain smoothness in the interior of . Received June 21, 1996 / In revised form December 2, 1996 / Accepted December 17, 1996     

Convex extensions and envelopes of lower semi-continuous functions

 We define a convex extension of a lower semi-continuous function to be a convex function that is identical to the given function over a pre-specified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions under which a convex extension can be constructed. When multiple convex extensions exist, we characterize the tightest convex extension in a well-defined sense. Using the notion of a generating set, we establish conditions under which the tightest convex extension is the convex envelope. Then, we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions. Finally, using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of $x/y$ over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outer-linearization of bilinear terms. Received: December 2000 / Accepted: May 2002 Published online: September 5, 2002 RID="*" ID="*" The research was funded in part by a Computational Science and Engineering Fellowship to M.T., and NSF CAREER award (DMI 95-02722) and NSF/Lucent Technologies Industrial Ecology Fellowship (NSF award BES 98-73586) to N.V.S. Key words. convex hulls and envelopes – multilinear functions – disjunctive programming – global optimization