Convex Potential Function Research Articles

A Large-Step Analytic Center Method for a Class of Smooth Convex Programming Problems

In this paper, a large-step analytic center method for smooth convex programming is proposed. The method is a natural implementation of the classical method of centers. It is assumed that the objective and constraint functions fulfil the so-called Relative Lipschitz Condition, with Lipschitz constant $M > 0$. A great advantage of the method, over the existing path-following methods, is that the steps can be made long by performing linesearches. In this method linesearches are performed along the Newton direction with respect to a strictly convex potential function when located far away from the central path. When sufficiently close to this path a lower bound for the optimal value is updated. It is proven that the number of iterations required by the algorithm to converge to an $\epsilon $-optimal solution is $O((1 + M^2 )\sqrt{n} | \ln \epsilon |)$ or $O((1 + M^2 )n | \ln \epsilon |)$, depending on the updating scheme for the lower bound.

A potential-reduction variant of Renegar's short-step path-following method for linear programming

We propose a new polynomial potential-reduction method for linear programming, which can also be seen as a large-step path-following method. We do an (approximate) linesearch along the Newton direction with respect to Renegar's strictly convex potential function if the iterate is far away from the central trajectory. If the iterate lies close to the trajectory, we update the lower bound for the optimal value. Dependent on this updating scheme, the iteration bound can be proved to be O( n L) or O( nL). Our method differs from the recently published potential-reduction methods in the choice of the potential function and the search direction.

Inverse problems and derivatives of determinants

This paper is about a well-posed and well-conditioned inverse problem. Instead of measurements «at the boundary», we go right inside a network-all nodes and edges are treated equally. By probing the network we find its response to m different source terms. From that information about the system, the goal is to find the conductances on the m edges. The equations are highly nonlinear. What is attractive is the appearance of a convex potential function, whose gradient is zero at the solution. The analysis of the problem becomes an analysis of this potential function