Related papers: A continuous mapping theorem for the smallest argm…
Herein, a methodology is developed to replicate functions, measures and stochastic processes onto a compact metric space. Many results are easily established for the replica objects and then transferred back to the original ones. Two…
Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors $v_1,\ldots,v_m \in \mathbb{R}^d$ and a constraint family ${\cal B}\subseteq 2^{[m]}$, find a set $S \in \cal{B}$ that…
We present algorithms for the Max-Cover and Max-Unique-Cover problems in the data stream model. The input to both problems are $m$ subsets of a universe of size $n$ and a value $k\in [m]$. In Max-Cover, the problem is to find a collection…
The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions.…
Consider finitely many nets of multivariate c\`adl\`ag stochastic processes. We show that the vectors consisting of the respective minimizing points converge in distribution to a random closed set. This set is given as a cartesian product…
The maximum principle is one of the most important tools in the analysis of geometric partial differential equations. Traditionally, the maximum principle is applied to a scalar function defined on a manifold, but in recent years more…
We consider the minimization or maximization of the $J$th largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi et al. (2014, SIAM J. Matrix Anal. Appl., 35, 699-724). This work addresses the setting…
The problem of computing the smallest fixed point of an order-preserving map arises in the study of zero-sum positive stochastic games. It also arises in static analysis of programs by abstract interpretation. In this context, the discount…
The concept of fixed point plays a crucial role in various fields of applied mathematics. The aim of this paper is to establish the existence of a unique fixed point of some type of functions which satisfy a new contraction principle,…
We study how maximal regularity estimates with respect to the continuous functions improve automatically in cases where the spatial norm is fundamentally different from the supremum norm. More precisely, we invoke properties such as weak…
We investigate the approximation of Monge--Kantorovich problems on general compact metric spaces, showing that optimal values, plans and maps can be effectively approximated via a fully discrete method. First we approximate optimal values…
Optimization algorithms appear in the core calculations of numerous Artificial Intelligence (AI) and Machine Learning methods, as well as Engineering and Business applications. Following recent works on the theoretical deficiencies of AI, a…
We give a new characterization of the space of functions of bounded variation in terms of a pointwise inequality connected to the maximal function of a measure. The characterization is new even in Euclidean spaces and it holds also in…
We present a new approach for finding a minimal value of an arbitrary function assuming only its continuity. The process avoids verifying Lagrange- or KKT-conditions. The method enables us to obtain a Brouwer fixed point (of a continuous…
We study the problem of maximizing the $k$-th eigenvalue functional over the class of absolutely continuous measures on a closed Riemannian manifold of dimension $m\geq 3$. For dimensions $3 \leq m \leq 6$, we generalize the work of…
In this paper, we develop a theory of new classes of discrete convex functions, called L-extendable functions and alternating L-convex functions, defined on the product of trees. We establish basic properties for optimization: a…
For a family of weight functions invariant under a finite reflection group, the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the…
This thesis investigates the design of algorithms for solving min-max optimization problems, which form the mathematical foundation of many modern applications in machine learning, game theory, and optimization. This work offers new…
We consider unconstrained multi-criteria optimization problems with finite sum objective functions. The proposed algorithm belongs to a non-monotone trust region framework where additional sampling approach is used to govern the sample size…
Recently, accelerated algorithms using the anchoring mechanism for minimax optimization and fixed-point problems have been proposed, and matching complexity lower bounds establish their optimality. In this work, we present the surprising…