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In this paper, we investigate optimization problems with nonnegative and orthogonal constraints, where any feasible matrix of size $n \times p$ exhibits a sparsity pattern such that each row accommodates at most one nonzero entry. Our…
Quadratic constrained quadratic programming problems often occur in various fields such as engineering practice, management science, and network communication. This article mainly studies a non convex quadratic programming problem with…
We consider the quadratic optimization problem $$F_n^{W,h}:= \sup_{x \in S^{n-1}} ( x^T W x/2 + h^T x )\,, $$ with $W$ a (random) matrix and $h$ a random external field. We study the probabilities of large deviation of $F_n^{W,h}$ for $h$ a…
We study the doubly nonnegative (DNN) relaxation of the standard quadratic optimization problem \[ \min\{x^\top Qx:\ x\in\Delta^{n-1}\},\qquad \Delta^{n-1}:=\{x\in\mathbb{R}_+^n:\ \mathbb{1}^\top x=1\}, \] for random symmetric matrices with…
This paper studies the lower bound complexity for the optimization problem whose objective function is the average of $n$ individual smooth convex functions. We consider the algorithm which gets access to gradient and proximal oracle for…
Montanari and Richard (2015) asked whether a natural semidefinite programming (SDP) relaxation can effectively optimize $\mathbf{x}^{\top}\mathbf{W} \mathbf{x}$ over $\|\mathbf{x}\| = 1$ with $x_i \geq 0$ for all coordinates $i$, where…
We propose a novel method that solves global optimization problems in two steps: (1) perform a (exponential) power-$N$ transformation to the not-necessarily differentiable objective function $f$ and get $f_N$, and (2) optimize the…
Smoothed analysis of multiobjective 0-1 linear optimization has drawn considerable attention recently. The number of Pareto-optimal solutions (i.e., solutions with the property that no other solution is at least as good in all the…
We study the oracle complexity of finding $\varepsilon$-Pareto stationary points in smooth multiobjective optimization with $m$ objectives. Progress is measured by the Pareto stationarity gap $\mathcal{G}(x)$, the norm of the best convex…
In this paper, a robust sequential quadratic programming method for constrained optimization is generalized to problem with an {expectation} objective function {and} deterministic equality and inequality constraints. A stochastic line…
The uniform quadratic optimizatin problem (UQ) is a nonconvex quadratic constrained quadratic programming (QCQP) sharing the same Hessian matrix. Based on the second-order cone programming (SOCP) relaxation, we establish a new sufficient…
This paper considers the problem of solving a special quartic-quadratic optimization problem with a single sphere constraint, namely, finding a global and local minimizer of…
Sparse principal component analysis with global support (SPCAgs), is the problem of finding the top-$r$ leading principal components such that all these principal components are linear combinations of a common subset of at most $k$…
Consider an optimization problem with $n$ binary variables and $d+1$ linear objective functions. Each valid solution $x \in \{0,1\}^n$ gives rise to an objective vector in $\R^{d+1}$, and one often wants to enumerate the Pareto optima among…
A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We…
This paper addresses black-box smooth optimization problems, where the objective and constraint functions are not explicitly known but can be queried. The main goal of this work is to generate a sequence of feasible points converging…
We prove that a "first-order" Sequential Quadratic Programming (SQP) algorithm for equality constrained optimization has local linear convergence with rate $(1-1/\kappa_R)^k$, where $\kappa_R$ is the condition number of the Riemannian…
A fundamental class of matrix optimization problems that arise in many areas of science and engineering is that of quadratic optimization with orthogonality constraints. Such problems can be solved using line-search methods on the Stiefel…
We consider a linear-quadratic optimization problem with pointwise bounds on the state for which the constraint is given by the Laplace-Beltrami equation (to have uniqueness we add an lower order term) on a two-dimensional surface . By…
A sequential quadratic programming method is designed for solving general smooth nonlinear stochastic optimization problems subject to expectation equality constraints. We consider the setting where the objective and constraint function…