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In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common…
We consider the problem of maximizing a monotone nondecreasing set function under multiple constraints, where the constraints are also characterized by monotone nondecreasing set functions. We propose two greedy algorithms to solve the…
Motivated by the need for, and growing interest in, modeling uncertainty in data, we introduce and study {\em stochastic minimum-norm optimization}. We have an underlying combinatorial optimization problem where the costs involved are {\em…
In this paper, we propose some new semidefinite relaxations for a class of nonconvex complex quadratic programming problems, which widely appear in the areas of signal processing and power system. By deriving new valid constraints to the…
In this article we study convex integer maximization problems with composite objective functions of the form $f(Wx)$, where $f$ is a convex function on $\R^d$ and $W$ is a $d\times n$ matrix with small or binary entries, over finite sets…
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
We study optimization algorithms for the finite sum problems frequently arising in machine learning applications. First, we propose novel variants of stochastic gradient descent with a variance reduction property that enables linear…
We propose new sequential simulation-optimization algorithms for general convex optimization via simulation problems with high-dimensional discrete decision space. The performance of each choice of discrete decision variables is evaluated…
Rank minimization is of interest in machine learning applications such as recommender systems and robust principal component analysis. Minimizing the convex relaxation to the rank minimization problem, the nuclear norm, is an effective…
Binary quadratic programming problems have attracted much attention in the last few decades due to their potential applications. This type of problems are NP-hard in general, and still considered a challenge in the design of efficient…
The main challenge of nonconvex optimization is to find a global optimum, or at least to avoid ``bad'' local minima and meaningless stationary points. We study here the extent to which algorithms, as opposed to optimization models and…
We study a convex resource allocation problem in which lower and upper bounds are imposed on partial sums of allocations. This model is linked to a large range of applications, including production planning, speed optimization, stratified…
We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying…
A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that…
The study of combinatorial optimization problems with a submodular objective has attracted much attention in recent years. Such problems are important in both theory and practice because their objective functions are very general. Obtaining…
In this paper, we study stochastic submodular maximization problems with general matroid constraints, that naturally arise in online learning, team formation, facility location, influence maximization, active learning and sensing objective…
Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this classic problem in the fully dynamic setting, where elements…
Majorization-minimization algorithms consist of successively minimizing a sequence of upper bounds of the objective function so that along the iterations the objective function decreases. Such a simple principle allows to solve a large…
Block majorization-minimization (BMM) is a simple iterative algorithm for constrained nonconvex optimization that sequentially minimizes majorizing surrogates of the objective function in each block while the others are held fixed. BMM…
We consider the densest submatrix problem, which seeks the submatrix of fixed size of a given binary matrix that contains the most nonzero entries. This problem is a natural generalization of fundamental problems in combinatorial…