Related papers: On Certain Conditions for Convex Optimization in H…
We present a fully iterative adaptive algorithm for the numerical minimization of strongly convex energy functionals in Hilbert spaces. The proposed approach, which we first present in abstract form, generates a hierarchical sequence of…
Predicting the value of a function $f$ at a new point given its values at old points is an ubiquitous scientific endeavor, somewhat less developed when $f$ produces multiple values that depend on one another, e.g. when it outputs…
In this paper, we consider optimization problems over closed embedded submanifolds of $\mathbb{R}^n$, which are defined by the constraints $c(x) = 0$. We propose a class of constraint dissolving approaches for these Riemannian optimization…
The study of convex functions - in particular, of their optimization (really minimization) is one of the most important fields of applied mathematics. Convexity seems to be one of those incredibly well-chosen hypotheses which is just…
The Douglas-Rachford algorithm is one of the most prominent splitting algorithms for solving convex optimization problems. Recently, the method has been successful in finding a generalized solution (provided that one exists) for…
We investigate a globalized inexact semismooth Newton method applied to strongly convex optimization problems in Hilbert spaces. Here, the semismooth Newton method is appplied to the dual problem, which has a continuously differentiable…
A quadratically constrained quadratic programming problem is considered in a Hilbert space setting, where neither the objective nor the constraint are convex functions. Necessary and sufficient conditions are provided to guarantee that the…
We study the query complexity of min-max optimization of a nonconvex-nonconcave function $f$ over $[0,1]^d \times [0,1]^d$. We show that, given oracle access to $f$ and to its gradient $\nabla f$, any algorithm that finds an…
We propose greedy and local search algorithms for rank-constrained convex optimization, namely solving $\underset{\mathrm{rank}(A)\leq r^*}{\min}\, R(A)$ given a convex function $R:\mathbb{R}^{m\times n}\rightarrow \mathbb{R}$ and a…
The divergence minimization problem plays an important role in various fields. In this note, we focus on differentiable and strictly convex divergences. For some minimization problems, we show the minimizer conditions and the uniqueness of…
We consider a class of convex optimization problems over the simplex of probability measures. Our framework comprises optimal experimental design (OED) problems, in which the measure over the design space indicates which experiments are…
Recently, a new local optimality concept for minimax problems, termed calm local minimax points, has been introduced. In this paper, we extend this concept to a general class of nonsmooth, nonconvex nonconcave minimax problems with coupled…
A new directional derivative and a new subdifferential for set-valued convex functions are constructed, and a set-valued version of the so-called 'max-formula' is proven. The new concepts are used to characterize solutions of convex…
We provide conditions ensuring that the KKT-type conditions characterizes the global optimality for quadratically constrained (possibly nonconvex) quadratic programming QCQP problems in Hilbert spaces. The key property is the convexity of a…
A major obstacle to non-convex optimization is the problem of getting stuck in local minima. We introduce a novel metaheuristic to handle this issue, creating an alternate Hamiltonian that shares minima with the original Hamiltonian only…
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 studies first-order algorithms for solving fully composite optimization problems over convex and compact sets. We leverage the structure of the objective by handling its differentiable and non-differentiable components…
Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone $K$, a norm $\|\cdot\|$ and a smooth convex function $f$, we want either 1) to minimize the norm over…
We consider the problem of choosing prices of a set of products so as to maximize profit, taking into account self-elasticity and cross-elasticity, subject to constraints on the prices. We show that this problem can be formulated as…
Motivated by recent work of Renegar, we present new computational methods and associated computational guarantees for solving convex optimization problems using first-order methods. Our problem of interest is the general convex optimization…