Related papers: On the Convex Interpolation for Linear Operators
Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced…
Autoencoders provide a powerful framework for learning compressed representations by encoding all of the information needed to reconstruct a data point in a latent code. In some cases, autoencoders can "interpolate": By decoding the convex…
Minimax optimization has been central in addressing various applications in machine learning, game theory, and control theory. Prior literature has thus far mainly focused on studying such problems in the continuous domain, e.g.,…
Functional constrained optimization is becoming more and more important in machine learning and operations research. Such problems have potential applications in risk-averse machine learning, semisupervised learning, and robust optimization…
We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization…
Most modern learning problems are over-parameterized, where the number of learnable parameters is much greater than the number of training data points. In this over-parameterized regime, the training loss typically has infinitely many…
Optimization problems involving minimization of a rank-one convex function over constraints modeling restrictions on the support of the decision variables emerge in various machine learning applications. These problems are often modeled…
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic…
We propose two novel conditional gradient-based methods for solving structured stochastic convex optimization problems with a large number of linear constraints. Instances of this template naturally arise from SDP-relaxations of…
Polynomial optimization encompasses a broad class of problems in which both the objective function and constraints are polynomial functions of the decision variables. In recent years, a substantial body of research has focused on…
We present and analyze an algorithm for optimizing smooth and convex or strongly convex objectives using minibatch stochastic gradient estimates. The algorithm is optimal with respect to its dependence on both the minibatch size and minimum…
Positive systems describing networks with inherently non-negative states and inputs arise naturally in routing, logistics, and compartmental modelling. We consider problems modelled as positive linear systems in incidence form with linear…
We examine the necessity of interpolation in overparameterized models, that is, when achieving optimal predictive risk in machine learning problems requires (nearly) interpolating the training data. In particular, we consider simple…
This paper deals with the optimization of Bolza problem with a system of convex and nonconvex, discrete and differential state variable inequality constraints of second order by deriving necessary and sufficient conditions for optimality.…
This paper considers mathematical programs, whose constraints are expressed by a parameterized vector equilibrium problem. The latter is a well recognized framework, which is able to cover multicriteria optimization, vector variational…
Necessary conditions for high-order optimality in smooth nonlinear constrained optimization are explored and their inherent intricacy discussed. A two-phase minimization algorithm is proposed which can achieve approximate first-, second-…
Composite optimization problems involve minimizing the composition of a smooth map with a convex function. Such objectives arise in numerous data science and signal processing applications, including phase retrieval, blind deconvolution,…
We introduce an interpolation--regression operator for polynomial approximation on the unit sphere $\mathbb{S}^2$ from discrete samples. The approximant is a spherical polynomial of degree $r$ which interpolates the data on a prescribed…
We study the problem of minimizing a convex function on a nonempty, finite subset of the integer lattice when the function cannot be evaluated at noninteger points. We propose a new underestimator that does not require access to…
This paper considers the inversion of ill-posed linear operators. To regularise the problem the solution is enforced to lie in a non-convex subset. Theoretical properties for the stable inversion are derived and an iterative algorithm akin…