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We study equilibrium models governed by parameter-dependent quasi-variational inequalities important from the viewpoint of optimization/equilibrium theory as well as numerous applications. The main focus is on quasi-variational inequalities with parameters entering both single-valued mad multivalued parts of the corresponding generalized equations in the sense of Robinson. The main tools of our variational analysis involve coderivatives of solution maps to quasi-variational inequalities, which allow us to obtain efficient conditions for robust Lipschitzian stability of quasi- variational inequalities and also to derive new necessary optimality conditions for mathematical programs with quasi-variational constraints. To conduct this analysis, we develop new results on coderivative calculus for structural settings involved in our models. The results obtained are illustrated by applications to some optimization and equilibrium models related to parameterized Nash games of two players and to oligopolistic market equilibria. [ABSTRACT FROM AUTHOR]

In this paper, by means of the concept of the working set, which is an estimate of the active set, we propose a feasible sequential linear equation algorithm for solving inequality constrained optimization problems. At each iteration of the proposed algorithm, we first solve one system of linear equations with a coefficient matrix of size m x m (where m is the number of constraints) to compute the working set; we then solve a subproblem which consists of four reduced systems of linear equations with a common coefficient matrix. Unlike existing QP-free algorithms, the subproblem is concerned with only the constraints corresponding to the working set. The constraints not in the working set are neglected. Consequently, the dimension of each subproblem is not of full dimension. Without assuming the isolatedness of the stationary points, we prove that every accumulation point of the sequence generated by the proposed algorithm is a KKT point, of the problem. Moreover, after finitely many iterations, the working set becomes independent of the iterates and is essentially the same as the active set of the KKT point. In other words, after finitely many steps, only those constraints which are active at the solution will be involved in the subproblem. Under some additional conditions, we show that the convergence rate is two-step superlinear or even Q-superlinear. We also report some preliminary numerical experiments to show that the proposed algorithm is practicable and effective for the test problems. [ABSTRACT FROM AUTHOR]

Iterative methods are proposed for certain augmented systems of linear equations that arise in interior methods for general nonlinear optimization, Interior methods define a sequence of KKT equations that represent the symmetrized (but indefinite) equations associated with Newton's method for a point satisfying the perturbed optimality conditions. These equations involve both the primal and dual variables and become increasingly ill-conditioned as the optimization proceeds. In this context, an iterative linear solver must not only handle the ill-conditioning but also detect the occurrence of KKT matrices with the wrong matrix inertia. A one-parameter family of equivalent linear equations is formulated that includes the KKT system as a special case. The discussion focuses on a particular system from this family, known as the "doubly augmented system," that is positive definite with respect to both the primal and dual variables. This property means that a standard preconditioned conjugate-gradient method involving both primal and dual variables will either terminate successfully or detect if the KKT matrix has the wrong inertia. Constraint preconditioning is a well-known technique for preconditioning the conjugate-gradient method on augmented systems. A family of constraint preconditioners is proposed that provably eliminates the inherent ill-conditioning in the augmented system. A considerable benefit of combining constraint preconditioning with the doubly augmented system is that the preconditioner need not be applied exactly. Two particular "active-set" constraint preconditioners are formulated that involve only a subset of the rows of the augmented system and thereby may be applied with considerably less work. Finally, some numerical experiments illustrate the numerical performance of the proposed preconditioners and highlight some theoretical properties of the preconditioned matrices. [ABSTRACT FROM AUTHOR]