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Journal of Convex Analysis 24 (2017), No. 4, 1117--1142
Copyright Heldermann Verlag 2017

Parametric Semidifferentiability of Minimax of Lagrangians: Averaged Adjoint Approach

Michel C. Delfour
Dép. de Mathématiques et de Statistique, Université de Montréal, CP 6128 / succ. Centre-ville, Montréal, Quebec, Canada H3C 3J7

Kevin Sturm
Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-str. 9, 45127 Essen, Germany

A standard approach to the minimization of a constrained objective function in the presence of equality constraints in Mathematical Programming or of a state equation in Control Theory is to introduce Lagrange multipliers or an adjoint state. The initial minimization problem is equivalent to the minimax of the associated Lagrangian.
In this paper, the targeted application is the computation of shape and topological derivatives (Sensitivity Analysis) where the control variable belongs to a metric space of geometries. In that context, it is natural to consider the right-hand side differentiability of the minimax of a Lagrangian with respect to a positive parameter as a first step towards the computation of semidifferentials and differentials on a metric space.
By using the new notion of averaged adjoint introduced by K. Sturm in two recent publications [On shape optimization with non-linear partial differential equations, Doctoral thesis, Technische Universität Berlin, Germany (2014); Minimax Lagrangian approach to the differentiability of non-linear PDE constrained shape functions without saddle point assumption, SIAM J. Control Optimization 53 (2015) 2017--2039], the minimax problem need not be related to a saddle point as in a paper by R. Correa and A. Seeger [Directional derivatives of a minimax function, Nonlinear Anal. Theory Methods Appl. 9 (1985) 13--22] and the so-called dual problem need not make sense. We extend Sturm's results from the single valued case to the case where the solutions of the state/averaged adjoint state equations are not unique. In such a case, non-differentiability can occur even if the functions at hand are infinitely differentiable as was illustrated by the seesaw problem in the seminal paper of J. M. Danskin [The theory of max-min, with applications, SIAM J. Applied Mathematics 14(4) (1966) 641--644].

Keywords: Minimax, Lagrangian, sensitivity analysis, semidifferential, averaged adjoint, shape, topological derivatives, optimal control, mathematical programming.

MSC: 49K20, 49K27, 49K35, 49K40, 49Q10, 49Q12.

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