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This paper investigates general and generalized differentiation properties of the optimal value function associated with perturbed optimization problems. Fundamental results on nearly convex sets and functions in infinite-dimensional spaces…

This paper investigates a specific class of nonsmooth nonconvex optimization problems in the face of data uncertainty, namely, robust optimization problems, where the given objective function can be expressed as a difference of two…

The convergence behavior of gradient methods for minimizing convex differentiable functions is one of the core questions in convex optimization. This paper shows that their well-known complexities can be achieved under conditions weaker…

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $S$. We assume that $S=C\cap A^{-1}(Q)$ is the nonempty solution set of a (multiple-set) split…

The performance of optimization methods is often tied to the spectrum of the objective Hessian. Yet, conventional assumptions, such as smoothness, do often not enable us to make finely-grained convergence statements -- particularly not for…

Many descent algorithms for multiobjective optimization have been developed in the last two decades. Tanabe et al. (Comput Optim Appl 72(2):339--361, 2019) proposed a proximal gradient method for multiobjective optimization, which can solve…

We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and…

We study Stochastic Gradient Descent (SGD) with diminishing step sizes for convex objective functions. We introduce a definitional framework and theory that defines and characterizes a core property, called curvature, of convex objective…

We consider simple bilevel optimization problems where the goal is to compute among the optimal solutions of a composite convex optimization problem, one that minimizes a secondary objective function. Our main contribution is threefold. (i)…

Typically, the sequence of points generated by an optimization algorithm may have multiple limit points. Under convexity assumptions, however, (sub)gradient methods are known to generate a convergent sequence of points. In this paper, we…

We study the ubiquitous super-resolution problem, in which one aims at localizing positive point sources in an image, blurred by the point spread function of the imaging device. To recover the point sources, we propose to solve a convex…

We propose an alternating subgradient method with non-constant step sizes for solving convex-concave saddle-point problems associated with general convex-concave functions. We assume that the sequence of our step sizes is not summable but…

We present a proximal gradient method for solving convex multiobjective optimization problems, where each objective function is the sum of two convex functions, with one assumed to be continuously differentiable. The algorithm incorporates…

Constrained Optimization solution algorithms are restricted to point based solutions. In practice, single or multiple objectives must be satisfied, wherein both the objective function and constraints can be non-convex resulting in multiple…

We consider a class of stochastic gradient optimization schemes. Assuming that the objective function is strongly convex, we prove weak error estimates which are uniform in time for the error between the solution of the numerical scheme,…

We present a composition rule involving quasiconvex functions that generalizes the classical composition rule for convex functions. This rule complements well-known rules for the curvature of quasiconvex functions under increasing functions…

Differential stability of convex discrete optimal control problems in Banach spaces is studied in this paper. By using some recent results of An and Yen [Appl. Anal. 94, 108--128 (2015)] on differential stability of parametric convex…

We consider a stationary variational inequality with gradient constraint and obstacle. We prove that this problem can be described by an equation using a Lagrange multiplier and a characteristic function. The Lagrange multiplier contains…

This paper explores a method for solving constrained optimization problems when the derivatives of the objective function are unavailable, while the derivatives of the constraints are known. We allow the objective and constraint function to…

Motivated by applications in optimization and machine learning, we consider stochastic quasi-Newton (SQN) methods for solving stochastic optimization problems. In the literature, the convergence analysis of these algorithms relies on strong…