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Optimal control of nonlinear evolution equations associated with time-dependent subdifferentials and applications

Noriaki Yamazaki (2009)

Banach Center Publications

In this paper we consider optimal control problems for abstract nonlinear evolution equations associated with time-dependent subdifferentials in a real Hilbert space. We prove the existence of an optimal control that minimizes the nonlinear cost functional. Also, we study approximating control problems of our equations. Then, we show the relationship between the original optimal control problem and the approximating ones. Moreover, we give some applications of our abstract results.

Optimality conditions for a class of mathematical programs with equilibrium constraints: strongly regular case

Jiří V. Outrata (1999)

Kybernetika

The paper deals with mathematical programs, where parameter-dependent nonlinear complementarity problems arise as side constraints. Using the generalized differential calculus for nonsmooth and set-valued mappings due to B. Mordukhovich, we compute the so-called coderivative of the map assigning the parameter the (set of) solutions to the respective complementarity problem. This enables, in particular, to derive useful 1st-order necessary optimality conditions, provided the complementarity problem...

Optimality conditions for an interval-valued vector problem

Ashish Kumar Prasad, Julie Khatri, Izhar Ahmad (2025)

Kybernetika

The present article considers a nonsmooth interval-valued vector optimization problem with inequality constraints. We first figure out Fritz John and Karush-Kuhn-Tucker type necessary optimality conditions for the interval-valued problem designed in the paper under quasidifferentiable 𝔉 -convexity in connection with compact convex sets. Subsequently, sufficient optimality conditions are extrapolated under aforesaid quasidifferentiability supported by a suitable numerical example.

Paraconvex functions and paraconvex sets

Huynh Van Ngai, Jean-Paul Penot (2008)

Studia Mathematica

We study a class of functions which contains both convex functions and differentiable functions whose derivatives are locally Lipschitzian or Hölderian. This class is a subclass of the class of approximately convex functions. It enjoys refined properties. We also introduce a class of sets whose associated distance functions are of that type. We discuss the properties of the metric projections on such sets under some assumptions on the geometry of the Banach spaces in which they are embedded. We...

Point derivations for Lipschitz functions andClarke's generalized derivative

Vladimir Demyanov, Diethard Pallaschke (1997)

Applicationes Mathematicae

Clarke’s generalized derivative f 0 ( x , v ) is studied as a function on the Banach algebra Lip(X,d) of bounded Lipschitz functions f defined on an open subset X of a normed vector space E. For fixed x X and fixed v E the function f 0 ( x , v ) is continuous and sublinear in f L i p ( X , d ) . It is shown that all linear functionals in the support set of this continuous sublinear function satisfy Leibniz’s product rule and are thus point derivations. A characterization of the support set in terms of point derivations is given.

Random perturbation of the variable metric method for unconstrained nonsmooth nonconvex optimization

Abdelkrim El Mouatasim, Rachid Ellaia, José Souza de Cursi (2006)

International Journal of Applied Mathematics and Computer Science

We consider the global optimization of a nonsmooth (nondifferentiable) nonconvex real function. We introduce a variable metric descent method adapted to nonsmooth situations, which is modified by the incorporation of suitable random perturbations. Convergence to a global minimum is established and a simple method for the generation of suitable perturbations is introduced. An algorithm is proposed and numerical results are presented, showing that the method is computationally effective and stable....

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