We extend the classic convergence rate theory for subgradient methods to apply to non-Lipschitz functions. For the deterministic projected subgradient method, we present a global O(1/ √ T ) convergence rate for any convex function which is locally Lipschitz around its minimizers. This approach is based on Shor’s classic subgradient analysis and implies generalizations of the standard convergenc...

We present an inexact interior point proximal method to solve linearly constrained convex problems. In fact, we derive a primal-dual algorithm to solve the KKT conditions of the optimization problem using a modified version of the rescaled proximal method. We also present a pure primal method. The proposed proximal method has as distinctive feature the possibility of allowing inexact inner step...

The subgradient method is used frequently to optimize dual functions in Lagrangian relaxation for separable integer programming problems. In the method, all subproblems must be solved optimally to obtain a subgradient direction. In this paper, the surrogate subgradient method is developed, where a proper direction can be obtained without solving optimally all the subproblems. In fact, only an a...

We present an inexact interior point proximal method to solve linearly constrained convex problems. In fact, we derive a primaldual algorithm to solve the KKT conditions of the optimization problem using a modified version of the rescaled proximal method. We also present a pure primal method. The proposed proximal method has as distinctive feature the possibility of allowing inexact inner steps...

We propose a modified extragradient method for solving the variational inequality problem in Hilbert space. The is combination of well-known subgradient with Mann’s mean value which updated iterate picked convex hull all previous iterates. show weak convergence to solution problem, provided that condition on corresponding averaging matrix fulfilled. Some numerical experiments are given effectiv...

In this note, we present a new averaging technique for the projected stochastic subgradient method. By using a weighted average with a weight of t + 1 for each iterate wt at iteration t, we obtain the convergence rate of O(1/t) with both an easy proof and an easy implementation. The new scheme is compared empirically to existing techniques, with similar performance behavior.

We derive a formula for the minimal time function where the dynamics are linear and the target is convex. Based on this formula, we give a new proof of the semiconvexity of the minimal time function, a result originally due to Cannarsa and Sinestrari.