This paper presents an application of canonical duality theory to the solution of contact problems with Coulomb friction. The contact problem is formulated as a quasi-variational inequality whose solution is found by solving its Karush– Kuhn–Tucker system of equations. The complementarity conditions are reformulated by using the Fischer–Burmeister complementarity function, obtaining a non-conve...

We extend the classic alternating direction method for convex optimization to solving the non-convex, nonnegative matrix factorization problem and conduct several carefully designed numerical experiments to compare the proposed algorithms with the most widely used two algorithms for solving this problem. In addition, the proposed algorithm is also briefly compared with two other more recent alg...

In this paper, we present a barrier method for solving nonlinear programming problems. It employs a Levenberg-Marquardt perturbation to the Karush-Kuhn-Tucker (KKT) matrix to handle indefinite Hessians and a line search to obtain sufficient descent at each iteration. We show that the Levenberg-Marquardt perturbation is equivalent to replacing the Newton step by a cubic regularization step with ...

For a locally optimal solution to the nonlinear semidefinite programming,under Robinson’s constraint qualification, we show that the nonsingularity of Clarke’sJacobian of the Fischer-Burmeister (FB) nonsmooth system is equivalent to the strongregularity of the Karush-Kuhn-Tucker point. Consequently, from Sun’s paper (Mathe-matics of Operations Research, vol. 31, pp. 761-776, 200...

When there is parameter uncertainty in the constraints of a convex optimization problem, it is natural to formulate the problem as a joint chance constrained program (JCCP) which requires that all constraints be satisfied simultaneously with a given large probability. In this paper, we propose to solve the JCCP by a sequence of convex approximations. We show that the solutions of the sequence o...

In this paper, we show that the moving directions of the primal-affine scaling method (with logarithmic barrier function), the dual-affine scaling method (with logarithmic barrier function), and the primal-dual interior point method are merely the Newton directions along three different algebraic "paths" that lead to a solution of the Karush-Kuhn-Tucker conditions of a given linear programming ...

In this work, we derive second-order optimality conditions for nonlinear semidefinite programming (NSDP) problems, by reformulating it as an ordinary nonlinear programming problem using squared slack variables. We first consider the correspondence between Karush-Kuhn-Tucker points and regularity conditions for the general NSDP and its reformulation via slack variables. Then, we obtain a pair of...

In this paper, we exploit the uplink radio resource allocation considering proportional fairness among users in heterogeneous wireless networks. Meanwhile, multi-mode user terminals are assumed to have the capability of using multiple radio access technologies simultaneously. To this end, a joint optimization problem is formulated for power and bandwidth allocation for uplink transmission. Sinc...

In the first part of the tutorial, we introduced the problem of unconstrained optimization, provided necessary and sufficient conditions for optimality of a solution to this problem, and described the gradient descent method for finding a (locally) optimal solution to a given unconstrained optimization problem. We now describe another method for unconstrained optimization, namely Newton’s metho...

The guaranteed cost control (GCC) problem involved in decentralized robust control of a class of uncertain nonlinear large-scale stochastic systems with high-order interconnections is considered. After determining the appropriate conditions for the stochastic GCC controller, a class of decentralized local state feedback controllers is derived using the linear matrix inequality (LMI). The extens...