Steepest Descent Methods with Generalized Distances for Constrained Optimization

We consider the problem s.t. , where C is a closed and covex subset of with nonempty interior, and introduce a family of interior point methods for this problem, which can be seen as approximate versions of generalized proximal point methods. Each step consists of a one-dimensional search along either a curve or a segment in the interior of C. The information about the boundary of C is contained in a generalized distance which defines the segment of the curve, and whose gradient diverges at the boundary of C. The objective of the search is either f or f plus a regularizing term. When , the usual steepest descent method is a particular case of our general scheme, and we manage to extend known convergence results for the steepest descent method to our family: for nonregularized one-dimensional searches,under a level set boundedness assumption on f, the sequence is bounded, the difference between consecutive iterates converges to 0 and every cluster point of the sequence satisfies first-order optimality conditions for the problem, i.e. is a solution if f is convex. For the regularized search and convex f, no boundedness condition on f is needed and full and global convergence of the sequence to a solution of the problem is established.