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In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework,…
The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved non convex…
We present two parallel optimization algorithms for a convex function $f$. The first algorithm optimizes over linear inequality constraints in a Hilbert space, $\mathbb H$, and the second over a non convex polyhedron in $\mathbb R^n$. The…
Error bounds, which refer to inequalities that bound the distance of vectors in a test set to a given set by a residual function, have proven to be extremely useful in analyzing the convergence rates of a host of iterative methods for…
The problem of minimizing a separable convex function under linearly coupled constraints arises from various application domains such as economic systems, distributed control, and network flow. The main challenge for solving this problem is…
We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation…
The multi-objective optimization is to optimize several objective functions over a common feasible set. Since the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies…
Sparsity and rank functions are important ways of regularizing under-determined linear systems. Optimization of the resulting formulations is made difficult since both these penalties are non-convex and discontinuous. The most common remedy…
We consider the problem of projecting a convex set onto a subspace, or equivalently formulated, the problem of computing a set obtained by applying a linear mapping to a convex feasible set. This includes the problem of approximating convex…
A multi-convex optimization problem is one in which the variables can be partitioned into sets over which the problem is convex when the other variables are fixed. Multi-convex problems are generally solved approximately using variations on…
We study a new penalty reformulation of constrained convex optimization based on the softplus penalty function. We develop novel and tight upper bounds on the objective value gap and the violation of constraints for the solutions to the…
We present an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions. The main motivator behind this numerical method is estimating solutions to Adverse Selection…
We focus on establishing the foundational paradigm of a novel optimization theory based on convolution with convex kernels. Our goal is to devise a morally deterministic model of locating the global optima of an arbitrary function, which is…
Why do neurons encode information the way they do? Normative answers to this question model neural activity as the solution to an optimisation problem; for example, the celebrated efficient coding hypothesis frames neural activity as the…
In this paper, we design a neural network architecture to approximate the weakly efficient frontier of convex vector optimization problems (CVOP) satisfying Slater's condition. The proposed machine learning methodology provides both an…
Analysis of the convergence rates of modern convex optimization algorithms can be achived through binary means: analysis of emperical convergence, or analysis of theoretical convergence. These two pathways of capturing information diverge…
Optimizing a nonlinear function over nonconvex sets is challenging since solving convex relaxations may lead to substantial relaxation gaps and infeasible solutions that must be "rounded" to feasible ones, often with uncontrollable losses…
In high-dimensional model selection problems, penalized simple least-square approaches have been extensively used. This paper addresses the question of both robustness and efficiency of penalized model selection methods, and proposes a…
Devising efficient algorithms that track the optimizers of continuously varying convex optimization problems is key in many applications. A possible strategy is to sample the time-varying problem at constant rate and solve the resulting…
We typically construct optimal designs based on a single objective function. To better capture the breadth of an experiment's goals, we could instead construct a multiple objective optimal design based on multiple objective functions. While…