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Unformatted text preview: Alternating minimization and projection methods for nonconvex problems 1 Hedy ATTOUCH 2 , J er ome BOLTE 3 , Patrick REDONT 2 , Antoine SOUBEYRAN 4 . Abstract We study the convergence properties of alternating proximal minimization algorithms for nonconvex structured functions of the following type: L ( x, y ) = f ( x ) + Q ( x, y ) + g ( y ) where f : R n R { + } and g : R m R { + } are proper lower semicontinuous functions and Q : R n R m R is a smooth C 1 (finite valued) function which couples the variables x and y . The algorithm is defined by: ( x , y ) R n R m given, ( x k , y k ) ( x k +1 , y k ) ( x k +1 , y k +1 ) 8 < : x k +1 argmin { L ( u, y k ) + 1 2 k k u x k k 2 : u R n } y k +1 argmin { L ( x k +1 , v ) + 1 2 k k v y k k 2 : v R m } Note that the above algorithm can be viewed as an alternating proximal minimization algorithm. Alternating projection algorithms on closed sets are particular cases of the above problem: just specialize f and g to be the indicator functions of the two sets and take Q ( x, y ) = k x y k 2 . The novelty of our approach is twofold: first, we work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka Lojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to metrically regular problems. Secondly, we rely on a new class of alternating minimization algorithms with costs to move which has recently been introduced by Attouch, Redont and Soubeyran, and which allows to handle general coupling functions Q . Our main result can be stated as follows: Assume that L has the Kurdyka Lojasiewicz property and that the sequence ( x k , y k ) k N is bounded. Then the trajectory has a finite length and, as a consequence, converges to a critical point of L . This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points (namely the Lojasiewicz exponent). As a striking application, we obtain the convergence of our alternating projection algorithm (a variant of the von Neumann algorithm) for a wide class of sets including in particular semialgebraic and tame sets, transverse smooth manifolds or sets with regular intersection. An illustration is given in the case of potential games, in which case the algorithm corresponds to a best response dynamic with inertia. The rate of convergence results allow to estimate the finite convergence time and the speed of convergence of the process to a Nash equilibrium. Key words Alternating minimization algorithms, alternating projections algorithms, proximal algorithms, non convex optimization, Kurdyka Lojasiewicz inequality, ominimal structures, tame optimization, convergence rate, finite convergence time, gradient systems, potential games, best response, inertia....
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 R.Srikant

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