334 research outputs found
Sample-path solutions for simulation optimization problems and stochastic variational inequalities
inequality;simulation;optimization
Proximal bundle method for contact shape optimization problem
From the mathematical point of view, the contact shape optimization is a problem of nonlinear optimization with a specific structure, which can be exploited in its solution. In this paper, we show how to overcome the difficulties related to the nonsmooth cost function by using the proximal bundle methods. We describe all steps of the solution, including linearization, construction of a descent direction, line search, stopping criterion, etc. To illustrate the performance of the presented algorithm, we solve a shape optimization problem associated with the discretized two-dimensional contact problem with Coulomb's friction
Approximate level method
In this paper we propose and analyze a variant of the level method [4], which is an algorithm for minimizing nonsmooth convex functions. The main work per iteration is spent on 1) minimizing a piecewise-linear model of the objective function and on 2) projecting onto the intersection of the feasible region and a polyhedron arising as a level set of the model. We show that by replacing exact computations in both cases by approximate computations, in relative scale, the theoretical iteration complexity increases only by the factor of four. This means that while spending less work on the subproblems, we are able to retain the good theoretical properties of the level method.evel method, approximate projections in relative scale, nonsmooth convex optimization, sensitivity analysis, large-scale optimization.
First-order methods of smooth convex optimization with inexact oracle
In this paper, we analyze different first-order methods of smooth convex optimization employing inexact first-order information. We introduce the notion of an approximate first-order oracle. The list of examples of such an oracle includes smoothing technique, Moreau-Yosida regularization, Modified Lagrangians, and many others. For different methods, we derive complexity estimates and study the dependence of the desired accuracy in the objective function and the accuracy of the oracle. It appears that in inexact case, the superiority of the fast gradient methods over the classical ones is not anymore absolute. Contrary to the simple gradient schemes, fast gradient methods necessarily suffer from accumulation of errors. Thus, the choice of the method depends both on desired accuracy and accuracy of the oracle. We present applications of our results to smooth convex-concave saddle point problems, to the analysis of Modified Lagrangians, to the prox-method, and some others.smooth convex optimization, first-order methods, inexact oracle, gradient methods, fast gradient methods, complexity bounds
Numerical methods for matching for teams and Wasserstein barycenters
Equilibrium multi-population matching (matching for teams) is a problem from
mathematical economics which is related to multi-marginal optimal transport. A
special but important case is the Wasserstein barycenter problem, which has
applications in image processing and statistics. Two algorithms are presented:
a linear programming algorithm and an efficient nonsmooth optimization
algorithm, which applies in the case of the Wasserstein barycenters. The
measures are approximated by discrete measures: convergence of the
approximation is proved. Numerical results are presented which illustrate the
efficiency of the algorithms.Comment: 29 pages, 13 figure
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