Related papers: SDP Achieves Exact Minimax Optimality in Phase Syn…
This work presents a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. The approach uses a particular weak embedding of the nonlinear PDE, resulting in a linear equation in the space…
We study population convergence guarantees of stochastic gradient descent (SGD) for smooth convex objectives in the interpolation regime, where the noise at optimum is zero or near zero. The behavior of the last iterate of SGD in this…
We study semidefinite programming (SDP) relaxations for the NP-hard problem of globally optimizing a quadratic function over the Stiefel manifold. We introduce a strengthened relaxation based on two recent ideas in the literature: (i) a…
Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover, a type of integer programming (IP) problem. A lattice-gas model on the Erd\"os-R\'enyi random graphs of $\alpha$-uniform…
Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate…
We consider a very wide class of models for sparse random Boolean 2CSPs; equivalently, degree-2 optimization problems over~$\{\pm 1\}^n$. For each model $\mathcal{M}$, we identify the "high-probability value"~$s^*_{\mathcal{M}}$ of the…
In this paper, we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly…
Distribution system state estimation (DSSE) is an essential tool for operation of distribution networks, the results of which enables the operator to have a thorough observation of the system. Thus, most distribution management systems…
Motivated by the task of clustering either $d$ variables or $d$ points into $K$ groups, we investigate efficient algorithms to solve the Peng-Wei (P-W) $K$-means semi-definite programming (SDP) relaxation. The P-W SDP has been shown in the…
Semidefinite programs (SDPs) are a fundamental class of optimization problems with important recent applications in approximation algorithms, quantum complexity, robust learning, algorithmic rounding, and adversarial deep learning. This…
We consider $\mathbb{Z}_2$-synchronization on the Euclidean lattice. Every vertex of $\mathbb{Z}^d$ is assigned an independent symmetric random sign $\theta_u$, and for every edge $(u,v)$ of the lattice, one observes the product…
We consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a…
With the rapid expansion in the scale of large language models (LLMs), enabling efficient distributed inference across multiple computing units has become increasingly critical. However, communication overheads from popular distributed…
We study the Max-Cut semidefinite programming (SDP) relaxation in the regime where a near-optimal solution admits a low-dimensional realization. While the Goemans--Williamson hyperplane rounding achieves the worst-case optimal approximation…
We study the convergence rates of the EM algorithm for learning two-component mixed linear regression under all regimes of signal-to-noise ratio (SNR). We resolve a long-standing question that many recent results have attempted to tackle:…
The stochastic block model (SBM) is a popular tool for community detection in networks, but fitting it by maximum likelihood (MLE) involves a computationally infeasible optimization problem. We propose a new semidefinite programming (SDP)…
We show that asymptotically, completely asynchronous stochastic gradient procedures achieve optimal (even to constant factors) convergence rates for the solution of convex optimization problems under nearly the same conditions required for…
A power system unit commitment (UC) problem considering uncertainties of renewable energy sources is investigated in this paper, through a distributionally robust optimization approach. We assume that the first and second order moments of…
Fast power system state estimation (SE) solution is of paramount importance for achieving real-time decision making in power grid operations. Semidefinite programming (SDP) reformulation has been shown effective to obtain the global optimum…
Convex relaxation methods are powerful tools for studying the lowest energy of many-body problems. By relaxing the representability conditions for marginals to a set of local constraints, along with a global semidefinite constraint, a…