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Error Bounds for Regularized ERROR BOUNDS FOR REGULARIZED COMPLEMENTARITY PROBLEMS y Paul Tseng July revised Septemb er Abstract An approach to solving a complementarity problem entails regular izingp erturb in g the problem by adding to the given mapping another mapping multiplied by a small p ositive parameter We study prop erties of the limit p oint of the solution to the regularized problem We also derive lo cal error b ounds on the distance from the solution to its limit p oint expressed in terms of the regularizatio n parameter Keywords Regularizati on complementarity problem optimization error b ound AMS sub ject classication M C C C C Intro duction n Consider the complementarity problem CP of nding an x satisfying T x F x F x x n n where F is a given continuous mapping This is a wellknown problem in optimization with many applications In various regulariza tioncontinuationsmoothing approaches to solving this problem one adds to n n the mapping F another mapping G multiplied by a small p ositive n scalar and computes p ossibly inexactly an x satisfying T x F x Gx F x Gx x Then one may decrease and up date x accordingly Our interests are in prop erties of any limit p oint of x along some sequence of and the distance from x to this limit p oint There are also the related issues of exis tenceuniquenessb oundednes s of x as which we will not fo cus on This research is supp orted by National Science Foundation Grant CCR y Department of Mathematics University of Washington Seattle Washington USA tsengmathwashingtonedu Paul Tseng The regularized CP is closely linked to a regularized smo oth optimization problem of the form minimize f u g u sub ject to u f u g u i m i i where f f f are continuously dierentiable functions dened on some m l op en set containing and g g g are continuously dierentiable func m l tions dened on m l In particular it is well known that the u asso ciated KarushKuhnTucker condition is exactly with x and v P P m m rg u v rg u rf u v rf u i i i i i i Gx F x m m g u f u i i i i Moreover if f f f are convex resp ectively quadratic on their resp ective m lm domains then this F is monotone resp ectively ane and continuous on Example and similarly for G A well studied case in this optimization setting is when g for i m ie constraint functions are unregularized i For the regularized CP one p opular choice of G is the identity mapping Gx x corresp onding to the wellknown Tikhonov regularization technique This choice has b een much studied including in the general setting of nding a zero of a maximal monotone op erator page Chapter I I The analogous choice of g u kuk for has b een considered by Karlin Mangasarian and others in the context of linear programs LP and by Tikhonov and various others in the general optimization setting see and references therein It was shown in Prop osition iii also see Theorem b Prop osition Theorem that if F is monotone then each limit p oint of x as is the least norm solution of CP Analogous results were obtained by Mangasarian in the context of LP also see for extensions to other choices of g in this context and by Levitin and Polyak and others in the general optimization setting see pages Prop osition and references therein If F is only a P function Sznader and Gowda Theorem showed that any limit p oint is weakParetominimal in the sense that no other solution is comp onentwise strictly less so any non p ositive solution is weakParetominimal A second p opular choice of G is the inverse function n Gx x j j corresp onding to logbarrier metho ds and interiorp oint metho ds This choice has b een considered Ko jima et al and Guler and in the Illp osed Variatonal Problems and Regularization Techniques general setting of nding a zero of a maximal monotone op erator by McLinden The analogous choice of l X lnu g u j j for has b een much studied in the context of LP see and references therein It was shown by McLinden Corollary that if F is monotone and CP has a strictly complementary solution then any limit p oint of x is a least weighted lnu solution of CP Analogus results were obtained by McLinden Theorem and Megiddo in the context of LP A third choice is the logarithm function n Gx lnx j j The analogous choice of l X u lnu g u j j j for was considered in the context of LP by Fang et al and from a dual exp onential p enalty view by Cominetti et al It was shown in that any limit p oint of the solution of the regularized LP is the least u lnuentropy solution of the LP This result was generalized recently by Auslender et al to convex programs with g b eing a certain kind of separable strictly convex essentially smo oth function A similar result was shown in for the LP case without the convexity and smo othness assumption Related results in the general optimization setting are given in and references therein These results do not assume g to b e separable or even continuous but they do need g to b e lower semicontinuous and realvalued at the limit p oint to b e meaningful As the preceding discussion shows there have b een many studies of the prop erties of a limit p ointx of x with particular fo cus on the cases of G given by or or However there have b een relatively few studies of the distance from x to x estimated in terms of In the context of LP with g given by this distance is known to b e in the order of e for some constant Theorem The same reference also gives distance estimates for the dual LP If g is more generally a separable strictly convex essentially smo oth function this distance can b e estimated in terms of rg and If the LP has a multicommo dity network ow structure and g is a weighted inverse barrier function whose weights are ane functions of a nonnegative variable p this distance is known to b e in the order of In this pap er we study the ab ove questions in the context of CP and its n regularization In particular we show that if F is pseudomonotone on and G is continuous atx thenx solves the variational inequality problem with Paul Tseng mapping G over the solution set Moreover if F is analytic on an op en set n T containing then the generalized distance Gx Gx x x is in the order of for some with if F is ane see Alternatively if n Gx G x j j j where each G is strictly increasing and continuous on but may tend to j at eg G given by or we show that in each co ordinate subspace over which F is pseudomonotone x solves the variational inequality problem with mapping G over the solution set see Prop osition a Moreover under the n assumption that either i F is pseudomonotone on and lim tG t t j for j J eg G given by or ii F is ane with certain principal submatrices of its Jacobian p ositive semidenite and spanning the corresp onding rows or iii F is ane with certain principal submatrix of its Jacobian p ositive semidenite and lim sup tG t for j J eg G given by we j t estimate jx x j in terms of x and in the case where F is ane we j j J j J j j estimate the latter in terms of where J is the set of indices j with G x j j see Prop osition cc Thus our results may b e applied to analyze regularization of a convex quadratic program of the form Our study is motivated by a related work in the context of LP although our results and our pro ofs are quite dierent from those in due to the dierent problem structure and regularization n In our notation denotes the space of ndimensional real column vectors n n n and denote the nonnegative orthant and the p ositive orthant in T n resp ectively and denotes transp ose For any x we denote by x the ith i comp onent of x and for any I N f ng by x the vector obtained I n by removing from x those x with i I and by x the vector in whose i I ith comp onent is x if i I and is zero otherwise Here and throughout i c means dene We denote by jI j the cardinality of I and denote I N nI p nn T x x kxk max jx j For any M and any I J N kxk iN i we denote by M the submatrix of M obtained by removing all rows of M with I indices outside of I and by M the submatrix of M obtained by removing IJ I n n all columns of M with indices outside of J For any F and any I n nonempty closed convex set we denote T VI F fx F x y x y g n Thus x satises if and only if x VI F and x satsies if and only n if x VI F G We denote by F the ith comp onent of F and for any i I N by F the mapping obtained by removing from F those F with i I I i n We say F is pseudomonotone on page if T T n and F y x y F x x y x y n n For any x and any nonempty closed set we denote dist x min ky xk y Illp osed Variatonal Problems and Regularization Techniques Error Bounds on Distance to Limiting Solu tion First we have the following b ound on the distance from x to the solution set of in terms of the regularization G x This is a simple consequence of an error b ound result for analytic systems n n Prop osition Consider an open set containing and an analytic n n n F with VI F nonempty Then for every bounded there exist and such that T distx k maxf G x gk jG x x j n n n for al l G and al l x VI F G Pro of We have that an x satises T x F x F x x n and that an x VI F G satises T T x F x G x F x x G x x So if x is also in the b ounded
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