This paper points out an important source of inefficiency in Smola and Schölkopf's sequential minimal optimization (SMO) algorithm for support vector machine (SVM) regression that is caused by the use of a single threshold value. Using clues from the KKT conditions for the dual problem, two threshold parameters are employed to derive modifications of SMO for regression. These modified algorithm...

This paper considers optimization problems with cardinality constraints. Based on a recently introduced reformulation of this problem as a nonlinear program with continuous variables, we first define some problem-tailored constraint qualifications and then show how these constraint qualifications can be used to obtain suitable optimality conditions for cardinality constrained problems. Here, th...

This paper proposes a solution algorithm to explicitly obtain the exact optimal solution of a second-order cone programming (SOCP) problem. The proposed solution algorithm is based on a parametric solution approach to determine the optimal strict region of parameters, and the main procedures are to perform deterministic equivalent transformations for the main SOCP problem and to solve the KKT c...

We introduce some approximate optimal solutions and a generalized augmented Lagrangian in nonlinear programming, establish dual function and dual problem based on the generalized augmented Lagrangian, obtain approximate KKT necessary optimality condition of the generalized augmented Lagrangian dual problem, prove that the approximate stationary points of generalized augmented Lagrangian problem...

This paper proposes a exact solution algorithm to explicitly obtain the exact optimal solution of a secondorder cone programming problem with box constraints of decision variables. The proposed solution algorithm is based on a parametric solution approach to determine the optimal strict region of parameters, and the main procedures are to perform deterministic equivalent transformations for the...

In this paper we consider a class of non-Lipschitz and non-convex minimization problems which generalize the L2 − Lp minimization problem. We propose an iterative algorithm that decides the next iteration based on the local convexity/concavity/sparsity of its current position. We show that our algorithm finds an -KKT point within O(log −1) iterations. The same result is also applied to the prob...

We propose a new set of preconditioners for the iterative solution, via a Preconditioned Conjugate Gradient (PCG) method, of the KKT systems that must be solved at each iteration of an Interior Point (IP) algorithm for the solution of linear Min Cost Flow (MCF) problems. These preconditioners are based on the idea of extracting a proper triangulated subgraph of the original graph which strictly...

Variational inequalities over sets deened by systems of equalities and inequalities are considered. A continuously diierentiable merit function is proposed whose unconstrained minima coincide with the KKT-points of the variational inequality. A detailed study of its properties is carried out showing that under mild assumptions this reformulation possesses many desirable features. A simple algor...

We address the iterative solution of symmetric KKT systems arising in the solution of convex quadratic programming problems. Two strictly related and well established formulations for such systems are studied with particular emphasis on the effect of preconditioning strategies on their relation. Constraint and augmented preconditioners are considered, and the choice of the augmentation matrix i...