44,156 research outputs found
Improved guarantees for optimal Nash equilibrium seeking and bilevel variational inequalities
We consider a class of hierarchical variational inequality (VI) problems that
subsumes VI-constrained optimization and several other important problem
classes including the optimal solution selection problem, the optimal Nash
equilibrium (NE) seeking problem, and the generalized NE seeking problem. Our
main contributions are threefold. (i) We consider bilevel VIs with merely
monotone and Lipschitz continuous mappings and devise a single-timescale
iteratively regularized extragradient method (IR-EG). We improve the existing
iteration complexity results for addressing both bilevel VI and VI-constrained
convex optimization problems. (ii) Under the strong monotonicity of the outer
level mapping, we develop a variant of IR-EG, called R-EG, and derive
significantly faster guarantees than those in (i). These results appear to be
new for both bilevel VIs and VI-constrained optimization. (iii) To our
knowledge, complexity guarantees for computing the optimal NE in nonconvex
settings do not exist. Motivated by this lacuna, we consider VI-constrained
nonconvex optimization problems and devise an inexactly-projected gradient
method, called IPR-EG, where the projection onto the unknown set of equilibria
is performed using R-EG with prescribed adaptive termination criterion and
regularization parameters. We obtain new complexity guarantees in terms of a
residual map and an infeasibility metric for computing a stationary point. We
validate the theoretical findings using preliminary numerical experiments for
computing the best and the worst Nash equilibria
Adaptive Preconditioned Gradient Descent with Energy
We propose an adaptive time step with energy for a large class of
preconditioned gradient descent methods, mainly applied to constrained
optimization problems. Our strategy relies on representing the usual descent
direction by the product of an energy variable and a transformed gradient, with
a preconditioning matrix, for example, to reflect the natural gradient induced
by the underlying metric in parameter space or to endow a projection operator
when linear equality constraints are present. We present theoretical results on
both unconditional stability and convergence rates for three respective classes
of objective functions. In addition, our numerical results shed light on the
excellent performance of the proposed method on several benchmark optimization
problems.Comment: 32 pages, 3 figure
Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry
Convex optimization is a well-established research area with applications in
almost all fields. Over the decades, multiple approaches have been proposed to
solve convex programs. The development of interior-point methods allowed
solving a more general set of convex programs known as semi-definite programs
and second-order cone programs. However, it has been established that these
methods are excessively slow for high dimensions, i.e., they suffer from the
curse of dimensionality. On the other hand, optimization algorithms on manifold
have shown great ability in finding solutions to nonconvex problems in
reasonable time. This paper is interested in solving a subset of convex
optimization using a different approach. The main idea behind Riemannian
optimization is to view the constrained optimization problem as an
unconstrained one over a restricted search space. The paper introduces three
manifolds to solve convex programs under particular box constraints. The
manifolds, called the doubly stochastic, symmetric and the definite multinomial
manifolds, generalize the simplex also known as the multinomial manifold. The
proposed manifolds and algorithms are well-adapted to solving convex programs
in which the variable of interest is a multidimensional probability
distribution function. Theoretical analysis and simulation results testify the
efficiency of the proposed method over state of the art methods. In particular,
they reveal that the proposed framework outperforms conventional generic and
specialized solvers, especially in high dimensions
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