On Continuity Properties of the Partial Legendre-fenchel Transform Convergence of Sequences of Augmented Lagrangian Functions, Moreau-yosida Approximates and Subdifferential Operators

In this article we consider the continuity properties of the partial Legendre-Fenchel transform which associates, with a bivariate convex function F: X x Y --t R U {+m}, its partial conjugate L : x x Y * + R, i.e. L(z , y* ) = inf y E Y { F ( z , y ) (Y* I Y)}. Follwiing [3] where this tranformation has been proved to be bicontinuous when convex functioiis F are equipped with the Mosco-epi-convergence, and convex concave Lagrangian functions L with the Mosco-epi/hypo-convergence, we now investigate the corresponding convergence notions for augmented Lagrangians, Moreau-Yosida approximates and subdifferential operators. 1. INTROD'CJCTION In 141, [5] the authors have introduced a new concept of convergence for bivariate functions specifically designed to study the convergence of sequences of saddle value problems, called epi/hypo-convergence. A main feature of this convergence notion is, in the convex setting, to make the partial Legendre-Fenchel transform bicontinuous. We recall that , given a convex function F: X x Y + R its partial Legendre-Fenchel transform is the convex-concave function L: x x Y * --* R The transformation F H L is one-to-one bicontinuous when convex functions are equipped with epi-convergence and closed convex-concave functions (in the sense of R.T. Rockafellar 1371 with epi/hypo convergence (see [5], [3])). When, following the classical duality scheme, functions F n are perturbation functions attached to the primal problems inf F n (5, 0) , %EX the above continuity property, combined with the variational properties of epi/hypo-convergence, is a key tool in order to study the convergence of the Partial Legendre-Fenchel transform 3 saddle points (that is of primal and dual solutions) of the corresponding Lagrangian functions {L" ; n E N}. The reduced problem is the study of epiconvergence of the sequence of perturbations functions {F" ; n E N}. This approach has been successfully applied to various situations in Convex Analysis (in Convex Programming see D. AzC [8], for convergence problems in Mechanics like homogenization of composite materials or reinforcement by thin structures see [9], H. Chabi [17], . . .). Indeed there are many other mathematical objects attached to this classical duality scheme. Our main purpose in this article is t o study for each of them the corresponding convergence notion. Particular attention is paid to the so-called augmented Lagrangian (especially quadratic augmented) whose definition is (compare with (1.1)) and which can be viewed as an "augmented "partial Legendre-Fenchel transform, In theorem 4.2 we prove the equivalence between Mosco epifhypoconvergence of Lagrangian functions L" and for every r > 0 and y* E Y *, the sequence of convex functions {Lp(. ,y*); n E N} (1.3) Mosco epi-converges to L r (a, y*). By the way, since Lr can be written as an inf-convolution we are led to study the two following basic properties of the inf-convolution operation, which explains the practical importance (especially from a numerical point of view) of the augmented Lagrangian : regularization effect; conservation of the infima and minimizing elements. This is considered in Propositions 3.1 and 3.2 for general convolution Kernels, see also M. Bougeard and J.P. Penot [14], M. Bougeard [13]. 4 H. Attouch, D. Azt! and R. Wets Iterating this regularization process but, now, on the %-variable, we obtain the so called MoreauYosida approximate the inf-sup being equal to the sup-inf (for closed convex-concave functions (theorem 5.1 d)) and the Mosco epi/hypo-convergence of Ln t o L is equivalent to the pointwise convergence of the associated Moreau-Yosida approximates (Theorem 5.2). Moreover L x , ~ has the same saddle elements as L! (Theorem 5.1 b). Finally we characterize in terms of graph convergence of subdifferential operators aLn -%aL the above notions (Theorem 6.1)’ and summarize in a diagram all these equivalent convergence properties. 2. CONVERGENCE OF CONVEX-CONCAVE SADDLE FUNCTIONS AND CONTINUITY OF THE PARTIAL LEGENDRE-FENCHEL TRANSFORMATION 2.1. Duality scheme Let us first briefly review the main feature of Rockafellar’s duality scheme (cf. [37], [38], [39]). Let X, Y , X*, Y * be linear spaces such that X (resp. Y ) is in separate duality with X * (resp. Y * ) via pairings denoted by (. I .). Let us consider L: x x Y * --+ R which is convex in the 5 variable, concave in the y* variable. Partial Legendre-Fenchel transform