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**11 - 18**of**18**### Necessary optimality conditions for optimal control problems with nonsmooth mixed state and control constraints

"... In this paper we study an optimal control problem with nonsmooth mixed state and control constraints. In most of the existing results, the necessary opti-mality condition for optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz condition and ..."

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In this paper we study an optimal control problem with nonsmooth mixed state and control constraints. In most of the existing results, the necessary opti-mality condition for optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz condition and under the assumption that the state and control constraint functions are smooth. In this paper we derive necessary optimality conditions for problems with nonsmooth mixed state and control constraints under constraint qualifications based on pseudo-Lipschitz continuity and calmness of certain set-valued maps. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold. Moreover necessary optimality conditions with an Euler inclusion taking an explicit multiplier form are derived for certain cases.

### DOI 10.1007/s10107-013-0667-7 FULL LENGTH PAPER Enhanced Karush–Kuhn–Tucker condition and weaker

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### Global Error

"... bounds for systems of convex polynomials over polyhedral constraints ..."

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### Serdica Math. J. 29 (2003), 301-314 APPLICATIONS OF THE FRÉCHET SUBDIFFERENTIAL

"... Abstract. In this paper we prove two results of nonsmooth analysis involving the Fréchet subdifferential. One of these results provides a necessary optimality condition for an optimization problem which arise naturally from a class of wide studied problems. In the second result we establish a suffic ..."

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Abstract. In this paper we prove two results of nonsmooth analysis involving the Fréchet subdifferential. One of these results provides a necessary optimality condition for an optimization problem which arise naturally from a class of wide studied problems. In the second result we establish a sufficient condition for the metric regularity of a set-valued map without continuity assumptions. 1. Preliminaries. Let X be a normed vector space and X ∗ its topological dual; we denote by BX, UX, SX the open unit ball, the closed unit ball and the unit sphere of X, respectively. By w and w ∗ we mean the weak topology on X and the weak star topology on X ∗. If S is a subset of X we denote by cl S the closure of S; if x ∈ X, we denote the distance from x to S by d(x,S) = infy∈S d(x,y) and by dS the distance function with respect to S,

### Necessary Optimality Conditions for Optimal Control Problems with Equilibrium Constraints

, 2015

"... Abstract. This paper introduces and studies the optimal control problem with equilib-rium constraints (OCPEC). The OCPEC is an optimal control problem with a mixed state and control equilibrium constraint formulated as a complementarity constraint and it can be seen as a dynamic mathematical program ..."

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Abstract. This paper introduces and studies the optimal control problem with equilib-rium constraints (OCPEC). The OCPEC is an optimal control problem with a mixed state and control equilibrium constraint formulated as a complementarity constraint and it can be seen as a dynamic mathematical program with equilibrium constraints (MPEC). It provides a powerful modeling paradigm for many practical problems such as bilevel optimal control problems and dynamic principal-agent problems. In this paper, we propose Fritz John type weak, Clarke, Mordukhovich and strong stationarities for the OCPEC. Moreover, we give some sufficient conditions to ensure that the local minimizers of the OCPEC are Fritz John type weakly stationary, Mordukhovich stationary and strongly stationary respectively. Key Words. Optimal control problem with equilibrium constraints, necessary optimality condition, weak stationarity, Mordukhovich stationarity, strong stationarity.

### Local cone approximations in optimization

, 2007

"... We show how to use intensively local cone approximations to obtain results in some fields of optimization theory such as optimality conditions, constraint qualifications, mean value theorems and error bound. ..."

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We show how to use intensively local cone approximations to obtain results in some fields of optimization theory such as optimality conditions, constraint qualifications, mean value theorems and error bound.

### Stability of error bounds for convex constraint systems in Banach spaces

, 2010

"... This paper studies stability of error bounds for convex constraint systems in Banach spaces. We show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one. A single ine ..."

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This paper studies stability of error bounds for convex constraint systems in Banach spaces. We show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one. A single inequality as well as semi-infinite constraint systems are considered.

### ERROR BOUNDS FOR VECTOR-VALUED FUNCTIONS ON METRIC SPACES

, 2011

"... In this paper, we attempt to extend the definition and existing local error bound criteria to vector-valued functions, or more generally, to functions taking values in a normed linear space. Some new primal space derivative-like objects – slopes – are introduced and a classification scheme of erro ..."

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In this paper, we attempt to extend the definition and existing local error bound criteria to vector-valued functions, or more generally, to functions taking values in a normed linear space. Some new primal space derivative-like objects – slopes – are introduced and a classification scheme of error bound criteria is presented.