Algorithm For Monotone Variational Inequalities Research Articles

Generalized Mirror Prox Algorithm for Monotone Variational Inequalities: Universality and Inexact Oracle

Generalized Mirror Prox Algorithm for Monotone Variational Inequalities: Universality and Inexact Oracle

Tight Ergodic Sublinear Convergence Rate of the Relaxed Proximal Point Algorithm for Monotone Variational Inequalities

Tight Ergodic Sublinear Convergence Rate of the Relaxed Proximal Point Algorithm for Monotone Variational Inequalities

Golden ratio algorithms for variational inequalities

The paper presents a fully adaptive algorithm for monotone variational inequalities. In each iteration the method uses two previous iterates for an approximation of the local Lipschitz constant without running a linesearch. Thus, every iteration of the method requires only one evaluation of a monotone operator F and a proximal mapping g. The operator F need not be Lipschitz continuous, which also makes the algorithm interesting in the area of composite minimization. The method exhibits an ergodic O(1 / k) convergence rate and R-linear rate under an error bound condition. We discuss possible applications of the method to fixed point problems as well as its different generalizations.

Distributionally robust project crashing with partial or no correlation information

AbstractCrashing is shortening the project makespan by reducing activity times in a project network by allocating resources to them. Activity durations are often uncertain and an exact probability distribution itself might be ambiguous. We study a class of distributionally robust project crashing problems where the objective is to optimize the first two marginal moments (means and SDs) of the activity durations to minimize the worst‐case expected makespan. Under partial correlation information and no correlation information, the problem is solvable in polynomial time as a semidefinite program and a second‐order cone program, respectively. However, solving semidefinite programs is challenging for large project networks. We exploit the structure of the distributionally robust formulation to reformulate a convex‐concave saddle point problem over the first two marginal moment variables and the arc criticality index variables. We then use a projection and contraction algorithm for monotone variational inequalities in conjunction with a gradient method to solve the saddle point problem enabling us to tackle large instances. Numerical results indicate that a manager who is faced with ambiguity in the distribution of activity durations has a greater incentive to invest resources in decreasing the variations rather than the means of the activity durations.

Weak convergence of a modified subgradient extragradient algorithm for monotone variational inequalities in Banach spaces

Weak convergence of a modified subgradient extragradient algorithm for monotone variational inequalities in Banach spaces

An explicit algorithm for monotone variational inequalities

We introduce a fully explicit method for solving monotone variational inequalities in Hilbert spaces, where orthogonal projections onto the feasible set are replaced by projections onto suitable hyperplanes. We prove weak convergence of the whole generated sequence to a solution of the problem, under only the assumptions of continuity and monotonicity of the operator and existence of solutions.

Comparison of two approximal proximal point algorithms for monotone variational inequalities

Proximal point algorithms (PPA) are attractive methods for solving monotone variational inequalities (MVI). Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approximate versions of PPA (APPA) are developed for practical applications. In this paper, we compare two APPA methods, both of which can be viewed as prediction-correction methods. The only difference is that they use different search directions in the correction-step. By extending the general forward-backward splitting methods, we obtain Algorithm I; in the same way, Algorithm II is proposed by spreading the general extra-gradient methods. Our analysis explains theoretically why Algorithm II usually outperforms Algorithm I.

Comparison of Two Proximal Point Algorithms for Monotone Variational Inequalities

Since proximal point algorithms (abbreviated as PPA) are attractive for solving monotone variational inequality problems, various approximate versions of PPA (APPA) are developed for practical applications. In this paper, we make a comparison between two different versions of APPAs (APPA I and APPA II) in the literature which share some common properties. Both of the algorithms use the same inexactness restriction and the same step length. The only difference is that they use different search directions. Through theoretical analysis and numerical experiment, we can see that APPA II usually performs better than APPA I.

Projection-splitting algorithms for monotone variational inequalities

We consider and analyze some new projection-splitting algorithms for solving monotone variational inequalities by using the technique of updating the solution. Our modification is in the spirit of the extragradient method. The modified methods converge for monotone continuous operators. The new iterative method differs from the existing projection methods.

On the convergence of descent methods for monotone variational inequalities

Recently, Zhu and Marcotte (1993) established the convergence of a modified descent algorithm for monotone variational inequalities. Using algorithmic equivalence results due to Patriksson (1993) and Larsson and Patriksson (1994), we show that this convergence result may be used to established the convergence of slightly modified versions of the classical successive approximation algorithms of Dafermos (1983) and Cohen (1988), and of the descent algorithms of Wu et al. (1993), Patriksson (1993) and Larsson and Patriksson (1994), under assumptions that are both much milder and much easier to verify than those for their original statements.