Globally Convergent Algorithms for Convex Programming

Abstract

We consider solving a (minimization) convex program by sequentially solving a (minimization) convex approximating subproblem and then executing a line search on an exact penalty function. Each subproblem is constructed from the current estimate of a solution of the given problem, possibly together with other information. Under mild conditions, solving the current subproblem generates a descent direction for the exact penalty function. Minimizing the exact penalty function along the current descent direction provides a new estimate of a solution, and a new subproblem is formed. For any arbitrary starting estimate, this scheme generates a sequence of estimates that converges to a solution of the given problem. Moreover, the functions defining the given problem and each subproblem need not be differentiable.

Disciplined Convex Programming

“…Several other classes of CPs have been identified recently as standard forms. These include semidefinite programs (SDPs) [155], second-order cone programs (SOCPs) [104], and geometric programs (GPs) [48,3,138,52,921. The work we present her applies to all of these special cases as well as to the general class of CPs.…”

Exact penalty method for minimizing of nonsmooth functions on convex sets

“…There are a lot of papers in which exact penalty functions are investigated, for example in [5]- [16].…”

On the Smoothing of the Square-Root Exact Penalty Function for Inequality Constrained Optimization

“…After Zangwill's development, extensive research of exact penalty function methods has been carried out in the literature (e.g, [2][3][4][5][6][7]). However, (2) is not a smooth function and causes some numerical instability problems in its implementation when the value of the penalty parameter ρ becomes larger.…”

“…The rest of this paper is organized as follows. In Section 2, we propose a method for smoothing the penalty function (3) in terms of first-order differentiability, which yields a first-order continuously differentiable penalty function. We prove some error estimates among the optimal objective function values of the smoothed penalty problem, of the nonsmooth penalty problem and of the original constrained optimization problem.…”

“…We present an algorithm to compute an approximate solution to (P) based on the smooth penalty function and show the convergence of the algorithm. In Section 3, we propose THE SMOOTHING OF THE SQUARE-ROOT EXACT PENALTY FUNCTION 377 another method for smoothing the penalty function (3) in terms of second-order differentiability, which yields a second-order continuously differentiable penalty function. We prove some error estimates and give an algorithm to compute an approximate solution to (P) based on the smooth penalty function.…”