相关论文: Localizing vector optimization problems with appli…
According to a result due to B.T. Polyak, a mapping between Hilbert spaces, which is $C^{1,1}$ around a regular point, carries a ball centered at that point to a convex set, provided that the radius of the ball is small enough. The present…
In the present paper, conditions under which the images of uniformly convex sets through $C^{1,1}$ regular mappings between Banach spaces remain convex are established. These conditions are expressed by a certain quantitative relation…
This paper considers distributed optimization for minimizing the average of local nonconvex cost functions, by using local information exchange over undirected communication networks. To reduce the required communication capacity, we…
This paper considers the distributed nonconvex optimization problem of minimizing a global cost function formed by a sum of local cost functions by using local information exchange. We first consider a distributed first-order primal-dual…
Motivated by recent increased interest in optimization algorithms for non-convex optimization in application to training deep neural networks and other optimization problems in data analysis, we give an overview of recent theoretical…
The main problem considered in the present paper is to single out classes of convex sets, whose convexity property is preserved under nonlinear smooth transformations. Extending an approach due to B.T. Polyak, the present study focusses on…
Vectorization is a technique that replaces a set-valued optimization problem with a vector optimization problem. In this work, by using an extension of Gerstewitz function [1], a vectorizing function is defined to replace a given set-valued…
In this paper, we establish the existence of the efficient solutions for polynomial vector optimization problems on a nonempty closed constraint set without any convexity and compactness assumptions. We first introduce the relative…
This paper provides characterizations of the weak solutions of optimization problems where a given vector function $F,$ from a decision space $X$ to an objective space $Y$, is "minimized" on the set of elements $x\in C$ (where $C\subset X$…
Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can…
In this work, we deal with unconstrained nonlinear optimization problems. Specifically, we are interested in methods carrying out updates possibly along directions not of descent, like Polyak's heavy-ball algorithm. Instead of enforcing…
Optimization of frame structures is formulated as a~non-convex optimization problem, which is currently solved to local optimality. In this contribution, we investigate four optimization approaches: (i) general non-linear optimization, (ii)…
We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has…
Lower-bound analyses for nonconvex strongly-concave minimax optimization problems have shown that stochastic first-order algorithms require at least $\mathcal{O}(\varepsilon^{-4})$ oracle complexity to find an $\varepsilon$-stationary…
Non-convex optimization is ubiquitous in modern machine learning. Researchers devise non-convex objective functions and optimize them using off-the-shelf optimizers such as stochastic gradient descent and its variants, which leverage the…
In this paper, we study a solution approach for set optimization problems with respect to the lower set less relation. This approach can serve as a base for numerically solving set optimization problems by using established solvers from…
In this paper, we are dealing with constrained vector optimisation problems where the objective function acts between real linear-topological spaces. Our aim is to study the relationships between the sets of properly efficient solutions to…
In the present paper, the following convexity principle is proved: any closed convex multifunction, which is metrically regular in a certain uniform sense near a given point, carries small balls centered at that point to convex sets, even…
This work presents a unified framework that combines global approximations with locally built models to handle challenging nonconvex and nonsmooth composite optimization problems, including cases involving extended real-valued functions. We…
An algorithm which computes a solution of a set optimization problem is provided. The graph of the objective map is assumed to be given by finitely many linear inequalities. A solution is understood to be a set of points in the domain…