Solving two-level variational inequality
Lex X be a non-empty, closed, convex subject of R{sup n} and G a continuous mapping from X into R{sup n}. Suppose that G is pseudo-monotone with respect to X and there exists a vector x{sup 0} {element_of} such that G(x{sup 0}) {element_of} int (O{sup +}X)*, where int({center_dot}) denotes the interior of the set. Here O{sup +}X is the recession cone of the set X, and C* is the dual cone of C {contained_in} R{sup n}. Under these assumptions, the following result obtains: The variational inequality (VI-problem): to find a vecto z {element_of} X such that (X - z){sup T}G(z) {>=} 0 {forall}x {element_of} X, has a non-empty, compact, convex solution set. Suppose further that problem solution set Z consists of more than one element, and consider the following VI-problem: to find vector z* {element_of} Z such that (z - z*){sup T}F(z*) {>=} 0 for all z {element_of} Z. Here, the mapping F:X {yields} R{sup n} is continuous and strictly monotone over X. Let us fix a positive parameter {epsilon} and consider the following parametrix VI-problem: to find a vector x{sup {epsilon}} {element_of} X such that (x - x{sub {epsilon}}){sup T}[G(x{sub {epsilon}}) + {epsilon}F(x{sub {epsilon}})] {>=} 0 {forall}x {element_of} X. If we assume that the mapping G is monotone over X, and keep intact all the above assumptions regarding G, F and Z, then the following result holds: For each sufficiently small value {epsilon} > 0, problem (4) has a unique solution x{sub {epsilon}}. Moreover, x{sub {epsilon}} converge to the solution z* of problem (2) - (3) when {epsilon} {yields} 0.
- OSTI ID:
- 36180
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
Similar Records
Global methods for optimizing over the efficient set
Asymptotic solution of the optimal control problem for standard systems with delay