1,069 research outputs found
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
An Interior-Point algorithm for Nonlinear Minimax Problems
We present a primal-dual interior-point method for constrained nonlinear, discrete minimax problems where the objective functions and constraints are not necessarily convex. The algorithm uses two merit functions to ensure progress toward the points satisfying the first-order optimality conditions of the original problem. Convergence properties are described and numerical results provided.Discrete min-max, Constrained nonlinear programming, Primal-dual interior-point methods, Stepsize strategies.
Randomized Lagrangian Stochastic Approximation for Large-Scale Constrained Stochastic Nash Games
In this paper, we consider stochastic monotone Nash games where each player's
strategy set is characterized by possibly a large number of explicit convex
constraint inequalities. Notably, the functional constraints of each player may
depend on the strategies of other players, allowing for capturing a subclass of
generalized Nash equilibrium problems (GNEP). While there is limited work that
provide guarantees for this class of stochastic GNEPs, even when the functional
constraints of the players are independent of each other, the majority of the
existing methods rely on employing projected stochastic approximation (SA)
methods. However, the projected SA methods perform poorly when the constraint
set is afflicted by the presence of a large number of possibly nonlinear
functional inequalities. Motivated by the absence of performance guarantees for
computing the Nash equilibrium in constrained stochastic monotone Nash games,
we develop a single timescale randomized Lagrangian multiplier stochastic
approximation method where in the primal space, we employ an SA scheme, and in
the dual space, we employ a randomized block-coordinate scheme where only a
randomly selected Lagrangian multiplier is updated. We show that our method
achieves a convergence rate of
for suitably defined
suboptimality and infeasibility metrics in a mean sense.Comment: The result of this paper has been presented at International
Conference on Continuous Optimization (ICCOPT) 2022 and East Coast
Optimization Meeting (ECOM) 202
- ā¦