Abstract
In this paper, the variational inequality with constraints can be viewed as an optimization problem. Using Lagrange function and projection operator, the equivalent operator equations for the variational inequality with constraints under the certain conditions are obtained. Then, the second-order differential equation system with the controlled process is established for solving the variational inequality with constraints. We prove that any accumulation point of the trajectory of the second-order differential equation system is a solution to the variational inequality with constraints. In the end, one example with three kinds of different cases by using this differential equation system is solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the controlled process for solving the variational inequality with constraints.
Highlights
We consider the variational inequality with constraints, denoted by VIP (K, F): find x∗ ∈ K such that〈F x∗, y − x∗〉 ≥ 0, ∀y ∈ K, (1)where K x ∈ Ω|g(x) ≤ 0, F: Rn ⟶ Rn is a monotone mapping, g: Rn ⟶ Rm is a convex and differentiable mapping, and Ω ⊆ Rn is a nonempty closed convex set.Variational inequality problems arise in physics, mechanics, economics, optimization, control, equilibrium model in transportation, and so forth. e finite dimensional variational inequality is an active field with abundant intension
Evtushenko and Zhadan have carried out a great deal of research on differential equation methods for nonlinear programming problems and constraint problems on general closed sets by using the stability theory of differential equations since 1973, which enriches the differential equation method of nonlinear programming problem. ere are other scholars who have done a lot of research work on differential equation methods, and they have established a variety of differential equation systems to solve the optimization problems
Since the nonlinear Lagrange function can be used to construct a dual algorithm for solving nonlinear optimization problems and the dual algorithm has no limitation on the feasibility of the original variables, many differential equation systems involved dual variables, such as those in the works of Zhou et al [15] and Jin et al [16]
Summary
We consider the variational inequality with constraints, denoted by VIP (K, F): find x∗ ∈ K such that. E variational inequality problem with coupled constraints and the fixed point of the extremal mapping with coupled constraints were studied by Antipin and Antipin [31, 32], where symmetric functions were introduced and the differential equation systems with the controlled process for global convergence were proposed. We will establish the second-order differential equation system with the controlled process for solving the variational inequality with constraints problem (1). When the function g is convex and differentiable, the relations in (6)–(14) are equivalent to each other
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