Related papers: Rank-one convexification for quadratic optimizatio…
This paper concerns with a noisy structured low-rank matrix recovery problem which can be modeled as a structured rank minimization problem. We reformulate this problem as a mathematical program with a generalized complementarity constraint…
Convexity prior is one of the main cue for human vision and shape completion with important applications in image processing, computer vision. This paper focuses on characterization methods for convex objects and applications in image…
We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a…
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…
Recent advances in the efficiency and robustness of algorithms solving convex quadratically constrained quadratic programming (QCQP) problems motivate developing techniques for creating convex quadratic relaxations that, although more…
Quadratic programming is a workhorse of modern nonlinear optimization, control, and data science. Although regularized methods offer convergence guarantees under minimal assumptions on the problem data, they can exhibit the slow…
Optimization problems with convex quadratic cost and polyhedral constraints are ubiquitous in signal processing, automatic control and decision-making. We consider here an enlarged problem class that allows to encode logical conditions and…
Nonlinear programming is explicitly analyzed via a novel perspective/method and from a bottom-up manner. The philosophy is based on the recent findings on convex quadratic equation (CQE), which help clarify a geometric interpretation that…
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization…
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…
Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this…
We study convex optimization problems where disjoint blocks of variables are controlled by binary indicator variables that are also subject to conditions, e.g., cardinality. Several classes of important examples can be formulated in such a…
The pooling problem is a classical NP-hard problem in the chemical process and petroleum industries. This problem is modeled as a nonlinear, nonconvex network flow problem in which raw materials with different specifications are blended in…
Non-convex optimization plays a key role in a growing number of machine learning applications. This motivates the identification of specialized structure that enables sharper theoretical analysis. One such identified structure is…
In the first part of this work [32], we introduce a convex parabolic relaxation for quadratically-constrained quadratic programs, along with a sequential penalized parabolic relaxation algorithm to recover near-optimal feasible solutions.…
In this work we study convex relaxations of quadratic optimisation problems over permutation matrices. While existing semidefinite programming approaches can achieve remarkably tight relaxations, they have the strong disadvantage that they…
This paper develops a convex approach for sparse one-dimensional deconvolution that improves upon L1-norm regularization, the standard convex approach. We propose a sparsity-inducing non-separable non-convex bivariate penalty function for…
This paper introduces mixed-integer optimization methods to solve regression problems that incorporate fairness metrics. We propose an exact formulation for training fair regression models. To tackle this computationally hard problem, we…
Variable selection is a fundamental task in statistical data analysis. Sparsity-inducing regularization methods are a popular class of methods that simultaneously perform variable selection and model estimation. The central problem is a…
A quadratically constrained quadratic program (QCQP) is an optimization problem in which the objective function is a quadratic function and the feasible region is defined by quadratic constraints. Solving non-convex QCQP to global…