Problems of optimal control are considered in the neoclassical Bolza format, which centers on states and velocities and relies on nonsmooth analysis. Subgradient versions of the EulerLagrange equation and the Hamiltonian equation are shown to be necessary for the optimality of a trajectory, moreover in a newly sharpened form that makes these conditions equivalent to each other. At the same time...

We propose a method for optimizing the lift-and-project relaxations of binary integer programs introduced by Lovász and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a restructuring of the relaxations, which isolates the complicating constraints and allows for a Lagrangian approach. We detail an enhanced subgradient method and discuss its efficient...

Determining the “active manifold” for a minimization problem is a large step towards solving the problem. Many researchers have studied under what conditions certain algorithms identify active manifolds in a finite number of iterations. In this work we outline a unifying framework encompassing many earlier results on identification via the Subgradient (Gradient) Projection Method, Newton-like M...

We discuss a bundle method to minimize non-smooth and non-convex locally Lipschitz functions. We analyze situations where only inexact subgradients or function values are available. For suitable classes of non-smooth functions we prove convergence of our algorithm to approximate critical points.

We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numeric...

In the networking research literature, the problem of network utility optimization is often converted to the dual problem which, due to nondifferentiability, is solved with a particular subgradient technique. This technique is not an ascent scheme, hence each iteration does not necessarily improve the value of the dual function. This paper examines the performance of this computational techniqu...

The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, we show that the sequential subgradient projecti...

In this paper we introduce an approximate optimization framework for solving graphs problems involving doubly stochastic matrices. This is achieved by using a low dimensional formulation of the matrices and the approximate solution is achieved by a simple subgradient method. We also describe one problem that can be solved using our method.

This paper considers a single-machine scheduling problem where the decision authorities and information are distributed in multiple sub-production systems. Sub-production systems share the single-machine and must cooperate with each other to achieve a global goal of minimizing a linear function of the completion times of the jobs; e.g., total weighted completion times. It is assumed that neithe...

In this paper, we propose a novel reduced-rank adaptive filtering algorithm by blending the idea of the Krylov subspace methods with the set-theoretic adaptive filtering framework. Unlike the existing Krylov-subspace-based reduced-rank methods, the proposed algorithm tracks the optimal point in the sense of minimizing the ‘true’ mean square error (MSE) in the Krylov subspace, even when the esti...