769 research outputs found
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.
Local stability and a renormalized Newton Method for equilibrium liquid crystal director modeling
We consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material. In the presence of pointwise unit-vector constraints and coupled electric fields, the numerical solution of such equations by Lagrange-Newton methods leads to problems with a double saddle-point form, for which we have previously proposed a preconditioned nullspace method as an effective solver [A. Ramage and E. C. Gartland, Jr., submitted]. The characterization of local stability of solutions is complicated by the double saddle-point structure, and here we develop efficiently computable criteria in terms of minimum eigenvalues of certain projected Schur complements. We also propose a modified outer iteration (“Renormalized Newton Method”) in which the orientation variables are normalized onto the constraint manifold at each iterative step. This scheme takes advantage of the special structure of these problems, and we prove that it is locally quadratically convergent. The Renormalized Newton Method bears some resemblance to the Truncated Newton Method of computational micromagnetics, and we compare and contrast the two
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
Frank-Wolfe Algorithms for Saddle Point Problems
We extend the Frank-Wolfe (FW) optimization algorithm to solve constrained
smooth convex-concave saddle point (SP) problems. Remarkably, the method only
requires access to linear minimization oracles. Leveraging recent advances in
FW optimization, we provide the first proof of convergence of a FW-type saddle
point solver over polytopes, thereby partially answering a 30 year-old
conjecture. We also survey other convergence results and highlight gaps in the
theoretical underpinnings of FW-style algorithms. Motivating applications
without known efficient alternatives are explored through structured prediction
with combinatorial penalties as well as games over matching polytopes involving
an exponential number of constraints.Comment: Appears in: Proceedings of the 20th International Conference on
Artificial Intelligence and Statistics (AISTATS 2017). 39 page
Optimization tools in management and finance
Through this paper, the most commonly used in management and finance optimization problems mathematical tools are presented, in a combined coherent way. The respective mathematical fundaments are synthetically outlined and the resolution methods are briefly described, hopping that this text functions as a manual in these matters.info:eu-repo/semantics/publishedVersio
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