An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities

摘要

Let C be a nonempty closed convex subset of a real Hilbert space (mathcal{H}) with inner product (langle cdot , cdot angle ), and let (f: mathcal{H}ightarrow mathcal{H}) be a nonlinear operator. Consider the inverse variational inequality (in short, (operatorname{IVI}(C,f))) problem of finding a point (xi ^{*}in mathcal{H}) such that $$ figl(xi ^{*}igr)in C, quad igllangle xi ^{*}, v-f igl(xi ^{*}igr)igrangle geq 0, quad orall vin C. $$ In this paper, we prove that (operatorname{IVI}(C,f)) has a unique solution if f is Lipschitz continuous and strongly monotone, which essentially improves the relevant result in (Luo and Yang in Optim. Lett. 8:1261a??1272, 2014). Based on this result, an iterative algorithm, named the alternating contraction projection method (ACPM), is proposed for solving Lipschitz continuous and strongly monotone inverse variational inequalities. The strong convergence of the ACPM is proved and the convergence rate estimate is obtained. Furthermore, for the case that the structure of C is very complex and the projection operator (P_{C}) is difficult to calculate, we introduce the alternating contraction relaxation projection method (ACRPM) and prove its strong convergence. Some numerical experiments are provided to show the practicability and effectiveness of our algorithms. Our results in this paper extend and improve the related existing results.KeywordsInverse variational inequality??Variational inequality??Lipschitz continuous??Strongly monotone??MSC47J20??90C25??90C30??90C52??1 IntroductionLet (mathcal{H}) be a real Hilbert space with inner product (langle cdot ,cdot angle ) and induced norm (Vert cdot Vert ). Recall that the metric projection operator of a nonempty closed convex subset C of (mathcal{H}), (P_{C}:mathcal{H}ightarrow C), is defined by $$ P_{C}(x):=rg min_{yin C} Vert x-y Vert ^{2}, quad xin mathcal{H}. $$Let C be a nonempty closed convex subset of (mathcal{H}), and let (F: Cightarrow mathcal{H}) be a nonlinear operator. The so-called variational inequality (in short, (operatorname{VI}(C,F))) problem is to find a point (u^{*}in C) such that $$ igllangle Figl(u^{*}igr),v-u^{*}igrangle geq 0, quad orall vin C. $$ (1) The variational inequalities have many important applications in different fields and have been studied intensively, see [1, 2, 4, 7, 8, 9, 10, 11, 12, 13, 14, 17, 24, 26, 27, 31, 32, 36, 38, 39, 40, 42, 43, 44, 45, 46], and the references therein.It is easy to verify that (u^{*}) solves (operatorname{VI}(C,F)) if and only if (u^{*}) is a solution of the fixed point equation $$ u^{*}=P_{C}(I-lambda F)u^{*}, $$ (2) where I is the identity operator on (mathcal{H}) and ?? is an arbitrary positive constant.A??class of variant variational inequalities is the inverse variational inequality (in short (operatorname{IVI}(C,f))) problem [19], which is to find a point (xi ^{*}in mathcal{H}) such that $$ figl(xi ^{*}igr)in C, quad igllangle xi ^{*}, v-figl(xi ^{*}igr)igrangle geq 0, quad orall vin C, $$ (3) where (f: mathcal{H}ightarrow mathcal{H}) is a nonlinear operator. The inverse variational inequalities are also widely used in many different fields such as the transportation system operation, control policies, and the electrical power network management [20, 22, 41].Now we give a brief overview of the properties and algorithms of inverse variational inequalities. For the properties of inverse variational inequalities, Han et al. [16] proved that the solution set of any monotone inverse variational inequality is convex. He [18] proved that the inverse variational inequality (operatorname{IVI}(C,f)) is equivalent to the following projection equation: $$ figl(xi ^{*}igr)=P_{C}igl(figl(xi ^{*} igr)-eta xi ^{*}igr), $$ where ?2 is an arbitrary positive constant. Consequently, the problem (operatorname{IVI}(C,f)) equals the fixed point problem of the mapping $$ T:=I-f+P_{C}(f-eta I). $$The following lemma reveals the intrinsic relationship between variational inequalities and inverse variational inequalities.