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We develop a Lagrange multiplier theory for nonconvex set-valued optimization problems under Lipschitz-type regularity conditions. Instead of classical continuous linear functionals, we introduce closed convex processes -- set-valued…
Learning processes are useful methodologies able to improve knowledge of real phenomena. These are often dependent on hyperparameters, variables set before the training process and regulating the learning procedure. Hyperparameters…
We study nonlinear optimization problems with a stochastic objective and deterministic equality and inequality constraints, which emerge in numerous applications including finance, manufacturing, power systems and, recently, deep neural…
We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schr\"odinger equations (NLS) on $\mathbb{R}^2$ with power nonlinearities of arbitrary odd degree. Specifically, the method in…
Optimality conditions in the form of a variational inequality are proved for a class of constrained optimal control problems of stochastic differential equations. The cost function and the inequality constraints are functions of the…
Many applications in science and engineering require the solution of large linear discrete ill-posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space-dimensions. The matrix…
In this paper we introduce the essential Lagrange multiplier and establish the solid mathematical foundation of constrained optimization in Hilbert spaces with sharp results on the mathematical foundation of quadratic-programming based…
An attempt is made to find a comprehensive mathematical framework in which to investigate the problems of well-posedness and asymptotic analysis for fully nonlinear evolutionary game theoretic models. The model should be rich enough to…
In the paper, we generalize the approach Gasnikov et. al, 2017, which allows to solve (stochastic) convex optimization problems with an inexact gradient-free oracle, to the convex-concave saddle-point problem. The proposed approach works,…
In this work, we consider a constrained convex problem with linear inequalities and provide an inexact penalty re-formulation of the problem. The novelty is in the choice of the penalty functions, which are smooth and can induce a non-zero…
We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is…
Large-scale constrained optimization is pivotal in modern scientific, engineering, and industrial computation, often involving complex systems with numerous variables and constraints. This paper provides a unified and comprehensive…
We study Tikhonov regularization for certain classes of non-linear ill-posed operator equations in Hilbert space. Emphasis is on the case where the solution smoothness fails to have a finite penalty value, as in the preceding study…
Motivated by infinite-dimensional optimal control problems with endpoint state constraints, in this Note, we introduce the notion of finite codimensional exact controllability for evolution equations. It is shown that this new…
We study a new penalty reformulation of constrained convex optimization based on the softplus penalty function. We develop novel and tight upper bounds on the objective value gap and the violation of constraints for the solutions to the…
We study the computational complexity certification of inexact gradient augmented Lagrangian methods for solving convex optimization problems with complicated constraints. We solve the augmented Lagrangian dual problem that arises from the…
This paper develops the proximal method of multipliers for a class of nonsmooth convex optimization. The method generates a sequence of minimization problems (subproblems). We show that the sequence of approximations to the solutions of the…
We consider the problem of computing optimal policies in average-reward Markov decision processes. This classical problem can be formulated as a linear program directly amenable to saddle-point optimization methods, albeit with a number of…
This paper studies a class of double-loop (inner-outer) algorithms for convex composite optimization. For unconstrained problems, we develop a restarted accelerated composite gradient method that attains the optimal first-order complexity…
We provide a generalization of first-order necessary conditions of optimality for infinite-dimensional optimization problems with a finite number of inequality constraints and with a finite number of inequality and equality constraints. Our…