A novel inexact smoothing method is presented for solving the second-order cone complementarity problems (SOCCP). Our method reformulates the SOCCP as an equivalent nonlinear system of equations by introducing a regularized Chen-Harker-Kanzow-Smale smoothing function. At each iteration, Newton’s method is adopted to solve the system of equations approximately, which saves computation work compa...

This paper investigates the potential behavior, both good and bad, of the well-known BFGS algorithm for smooth minimization, when applied to nonsmooth functions. We consider three very particular examples. We first present a simple nonsmooth example, illustrating how BFGS (in this case with an exact line search) typically succeeds despite nonsmoothness. We then study, computationally, the behav...

In this paper we present an extension to SDP of the well known infeasible Interior Point method for linear programming of Kojima, Megiddo and Mizuno (A primal-dual infeasibleinterior-point algorithm for Linear Programming, Math. Progr., 1993). The extension developed here allows the use of inexact search directions; i.e., the linear systems defining the search directions can be solved with an a...

In this paper we present an extension to SDP of the well known infeasible Interior Point method for linear programming of Kojima, Megiddo and Mizuno (A primal-dual infeasibleinterior-point algorithm for Linear Programming, Math. Progr., 1993). The extension developed here allows the use of inexact search directions; i.e., the linear systems defining the search directions can be solved with an a...

We study inexact proximal Newton-type methods to solve convex optimization problems in composite form: minimize x∈Rn f(x) := g(x) + h(x), where g is convex and continuously differentiable and h : R → R is a convex but not necessarily differentiable function whose proximal mapping can be evaluated efficiently. Proximal Newton-type methods require the solution of subproblems to obtain the search ...

This tutorial describes the classic method of conjugate directions: the generalization of the conjugate-gradient method in iterative least-square inversion. I derive the algebraic equations of the conjugate-direction method from general optimization principles. The derivation explains the “magic” properties of conjugate gradients. It also justifies the use of conjugate directions in cases when ...

Convex quadratic semidefinite programming (QSDP) has been widely applied in solving engineering and scientific problems such as nearest correlation problems and nearest Euclidean distance matrix problems. In this paper, we study an inexact primal-dual infeasible path-following algorithm for QSDP problems of the form: minX{12X • Q(X) + C •X : A(X) = b, X 0}, where Q is a self-adjoint positive se...

A wide variety of problems involving analysis of systems can be rewritten as a semidefinite program. When solving these problems optimization algorithms are used. Large size makes the problems unsolvable in practice and computationally more effective solvers are needed. This paper investigates how to exploit structure and problem knowledge in some control applications. It is shown that inexact ...

In this paper we present a convergence analysis for some inexact (polynomial) variants of the infeasible-interior-point-algorithm of Kojima, Megiddo and Mizuno. For this analysis we assume that the iterates are bounded. The new variants allow the use of search directions that are calculated from the deening linear system with only moderate accuracy, e.g. via the use of Krylov subspace methods l...

In this paper we study a class of derivative-free unconstrained minimization algorithms employing nonmonotone inexact linesearch techniques along a set of suitable search directions. In particular, we define globally convergent nonmonotone versions of some well-known derivativefree methods and we propose a new algorithm combining coordinate rotations with approximate simplex gradients. Through ...