First-order approximations for differential inclusions

It is shown, under a mere continuity assumption, that the union of affine functions generated by the right-hand side of a differential inclusion, is a little oh approximation of the attainable set. Explicit estimates are given. An application to polygonal approximations is displayed.Research supported by a grant from the Basic Research Fund, The Israel Academy of Science and Humanities.Incumbent of the Hettie H. Heineman Professorial Chair in Mathematics.     

Facial reduction in partially finite convex programming

We consider the problem of minimizing an extended-valued convex function on a locally convex space subject to a finite number of linear (in)equalities. When the standard constraint qualification fails a reduction technique is needed to derive necessary optimality conditions. Facial reduction is usually applied in the range of the constraints. In this paper it is applied in the domain space, thus maintaining any structure (and in particular lattice properties) of the underlying domain. Applications include constrained approximation and best entropy estimation.Research partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Nonsmooth generalized guiding functions for periodic differential inclusions

We consider the periodic problem for differential inclusions in $$ \user2{\mathbb{R}}^{\rm N} $$ with a nonconvex-valued orientor field F(t, ζ), which is lower semicontinuous in $$ \zeta \in \user2{\mathbb{R}}^{\rm N} $$ Using the notion of a nonsmooth, locally Lipschitz generalized guiding function, we prove that the inclusion has periodic solutions. We have two such existence theorems. We also study the “convex” periodic problem and prove an existence result under upper semicontinuity hypothesis on F(t, ·) and using a nonsmooth guiding function. Our work was motivated by the recent paper of Mawhin-Ward [23] and extends the single-valued results of Mawhin [19] and the multivalued results of De Blasi-Górniewicz-Pianigiani [4], where either the guiding function is C¹ or the conditions on F are more restrictive and more difficult to verify.     

Periodic solutions to impulsive differential inclusions with constraints

The existence of a periodic solution to an impulsive differential inclusion being invariant with respect to a non-convex set of state constraints is established by the use a Lefschetz type fixed-point theorem for set-valued maps.     

Singularly perturbed differential inclusions: An averaging approach

We consider nonlinear, singularly perturbed differential inclusions and apply the averaging method in order to construct a limit differential inclusion for slow motion. The main approximation result states that the existence and regularity of the limit differential inclusion suffice to describe the limit behavior of the slow motion. We give explicit approximation rates for the uniform convergence on compact time intervals. The approach works under controllability or stability properties of fast motion.     

Singularly perturbed functional-differential inclusions

Differential inclusions of a retarded type with a small real parameter >0 in part of the derivatives are considered. We prove upper semicontinuity of the map set of solutions at =0⁺ inC[0, 1]×(L ²(0, 1)–weak) topology. In case of constant delay lower semicontinuity inC[0, 1]×(L ¹(0, 1)–strong) is shown.     

Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming

For an inequality system defined by an infinite family of proper convex functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications. Under the new constraint qualifications, we provide necessary and/or sufficient conditions for the KKT rules to hold. Similarly, we provide characterizations for constrained minimization problems to have total Lagrangian dualities. Several known results in the conic programming problem are extended and improved.     

Solvability of strongly nonlinear boundary value problems for second order differential inclusions

In this paper we study a problem for a second order differential inclusion with Dirichlet, Neumann and mixed boundary conditions. The equation is driven by a nonlinear, not necessarily homogeneous, differential operator satisfying certain conditions and containing, as a particular case, the p$p$-Laplacian operator. We prove the existence of solutions both for the case in which the multivalued nonlinearity has convex values and for the case in which it has not convex values. The presence of a maximal monotone operator in the equation make the results applicable to gradient systems with non-smooth, time invariant, convex potential and differential variational inequalities.     

Impulsive semilinear differential inclusions: Topological structure of the solution set and solutions on non-compact domains

This paper deals with an impulsive Cauchy problem governed by the semilinear evolution differential inclusion x^′(t)∈A(t)x(t)+F(t,x(t))$x^{\prime }\left(t\right)\in A\left(t\right)x\left(t\right)+F(t,x\left(t\right))$, where _{{A(t)}t∈[0,b]}${\left\{A\left(t\right)\right\}}_{t\in [0,b]}$ is a family of linear operators (not necessarily bounded) in a Banach space E$E$ generating an evolution operator and F$F$ is a Carathéodory type multifunction. First a theorem on the compactness of the set of all mild solutions for the problem is given. Then this result is applied to obtain the existence of mild solutions for the impulsive Cauchy problem defined on non-compact domains.     

Partially finite convex programming,Part II: Explicit lattice models

In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion ofquasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming,L ¹ approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation.     

On the continuity of midpoint hull convex set-valued functions

Summary The concept of hull convexity (midpoint hull convexity) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of convexity (midpoint convexity).The main result is a sufficient condition for a midpoint hull convex set-valued function to be continuous. This theorem improves a result obtained by K. Nikodem (Bulletin of the Polish Academy of Sciences, Mathematics,34 (1986), 393–399).     

Convergence of decreasing sequences of convex sets in nonreflexive Banach spaces

We consider nested sequences of linear or convex closed sets of the form arising in estimation and other inverse problems. We show that such sequences may fail to converge in any of the recently studied set convergences other than Mosco convergence. We also provide a positive result concerning the epislice convergence of related sequences of functions.Research partially supported by NSERC operating grants.     

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.     

A general multi-valued hybrid fixed point theorem and perturbed differential inclusions

Combining three basic multi-valued versions of Banach, Schauder and Tarski fixed point theorems, a general hybrid fixed point theorem for multi-valued mappings in Banach spaces is proved via measure of noncompactness and it is further applied to perturbed differential inclusions for proving the existence results under mixed Lipschitz, compactness and monotone conditions.     

Convex solution of a permutation problem

In this paper, we show that a problem of finding a permuted version of k vectors from R^N such that they belong to a prescribed rank r subset, can be solved by convex optimization. We prove that under certain generic conditions, the wanted permutation matrix is unique in the convex set of doubly-stochastic matrices. In particular, this implies a solution of the classical correspondence problem of finding a permutation that transforms one collection of points in R^k into the another one. Solutions to these problems have a wide set of applications in Engineering and Computer Science.     

Systems of first order differential inclusions with maximal monotone terms

In this paper, we establish the existence of solutions to systems of first order differential inclusions with maximal monotone terms satisfying the periodic boundary condition. Our proofs rely on the theory of maximal monotone operators, and the Schauder and the Kakutani fixed point theorems. A notion of solution-tube to these problems is introduced. This notion generalizes the notion of upper and lower solutions of first order differential equations.     

Continuous selection of the solution map for one-sided Lipschitz differential inclusions

In the paper we deal with differential inclusions with one-sided Lipschitz (OSL) continuous right-hand sides, and prove the existence of a continuous selection of the solution map which assigns to any point the set of solutions to the multivalued Cauchy problem. As an application we study the problem of the existence of viable trajectories in a prescribed closed subset of a Banach space via the generalized Wa?ewski retract method.