Quoc Tran Dinh

We study the computational complexity certification of inexact gradient augmented Lagrangian methods for solving convex optimization problems with complicated constraints. We solve the augmented Lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information. Moreover, since the exact solution of the augmented Lagrangian primal problem is hard to compute in practice, we solve this problem up to some given inner accuracy. We derive relations between the inner and the outer accuracy of the primal and dual problems and we give a full convergence rate analysis for both gradient and fast gradient algorithms. We provide estimates on the primal and dual suboptimality and on primal feasibility violation of the generated approximate primal and dual solutions. Our analysis relies on the Lipschitz property of the dual function and on inexact dual gradients. We also discuss implementation aspects of the proposed algorithms on constrained model predictive control problems for embedded linear systems.

In this work, we propose a new local optimization method to solve a class of nonconvex semidefinite programming (SDP) problems. The basic idea is to approximate the feasible set of the nonconvex SDP problem by inner positive semidefinite convex approximations via a parameterization technique. This leads to an iterative procedure to search a local optimum of the nonconvex problem. The convergence of the algorithm is analyzed under mild assumptions. Applications in static output feedback control are benchmarked and numerical tests are implemented based on the data from the COMPLeib library.

Unlike convex case, a local equilibrium point of a nonconvex Nash-Cournot oligopolistic equilibrium problem may not be a global one. Finding such a local equilibrium point or even a stationary point of this problem is not an easy task. This paper deals with a numerical method for Nash-Cournot equilibrium models involving nonconvex cost functions. We develop a local method to compute a stationary point of this class of problems. The convergence of the algorithm is proved and its complexity is estimated under certain assumptions. Numerical examples are implemented to illustrate the convergence behavior of the proposed algorithm.