In this paper, we approach the problem of finding the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space via an implicit forward-backward-forward dynamical system with nonconstant relaxation parameters and stepsizes of the resolvents. Besides proving existence and uniqueness of strong global solutions … Read more
We consider linear generalized Nash games and introduce the so-called cone condition which characterizes the smoothness of the Nikaido-Isoda function under weak assumptions. The latter mapping arises from a reformulation of the generalized Nash equilibrium problem as a possibly nonsmooth optimization problem. Other regularity conditions like LICQ or SMFC(Q) are only sufficient for smoothness, but … Read more
Large scale nonsmooth convex optimization is a common problem for a range of computational areas including machine learning and computer vision. Problems in these areas contain special domain structures and characteristics. Special treatment of such problem domains, exploiting their structures, can significantly improve the the computational burden. We present a weighted Mirror Descent method to … Read more
Recently, optimization problems involving nonsmooth and locally Lipschitz functions have been subject of investigation, and an innovative method known as Gradient Sampling has gained attention. Although the method has shown good results for important real problems, some drawbacks still remain unexplored. This study suggests modifications to the gradient sampling class of methods in order to … Read more
We begin by considering second order dynamical systems of the from $\ddot x(t) + \Gamma (\dot x(t)) + \lambda(t)B(x(t))=0$, where $\Gamma: {\cal H}\rightarrow{\cal H}$ is an elliptic bounded self-adjoint linear operator defined on a real Hilbert space ${\cal H}$, $B: {\cal H}\rightarrow{\cal H}$ is a cocoercive operator and $\lambda:[0,+\infty)\rightarrow [0,+\infty)$ is a relaxation function depending … Read more
In this paper, we consider a class of Forward–Backward (FB) splitting methods that includes several variants (e.g. inertial schemes, FISTA) for minimizing the sum of two proper convex and lower semi-continuous functions, one of which has a Lipschitz continuous gradient, and the other is partly smooth relatively to a smooth active manifold $\mathcal{M}$. We propose … Read more
In this paper we present two inexact proximal point algorithms to solve minimization problems for quasiconvex objective functions on Hadamard manifolds. We prove that under natural assumptions the sequence generated by the algorithms are well defined and converge to critical points of the problem. We also present an application of the method to demand theory … Read more
We describe a new possibility to model disjunctive optimization problems as generalized semi-infinite programs. In contrast to existing methods, for our approach neither a conjunctive nor a disjunctive normal form is expected. Applying existing lower level reformulations for the corresponding semi-infinite program we derive conjunctive nonlinear problems without any logical expressions, which can be locally … Read more
In this paper, we consider a class of constrained optimization problems where the feasible set is a general closed convex set and the objective function has a nonsmooth, nonconvex regularizer. Such regularizer includes widely used SCAD, MCP, logistic, fraction, hard thresholding and non-Lipschitz $L_p$ penalties as special cases. Using the theory of the generalized directional … Read more
We develop an optimization technique to compute local solutions to synthesis problems subject to integral quadratic constraints (IQCs). We use the fact that IQCs may be transformed into semi-infinite maximum eigenvalue constraints over the frequency axis and approach them via nonsmooth optimization methods. We develop a suitable spectral bundle method and prove its convergence in … Read more