Related papers: Exploiting symmetries in SDP-relaxations for polyn…
This paper addresses the problem of solving a class of nonlinear optimal control problems (OCP) with infinite-dimensional linear state constraints involving Riesz-spectral operators. Each instance within this class has time/control…
The max-cut problem is a classical graph theory problem which is NP-complete. The best polynomial time approximation scheme relies on \emph{semidefinite programming} (SDP). We study the conditions under which graphs of certain classes have…
We study graph clustering in the Stochastic Block Model (SBM) in the presence of both large clusters and small, unrecoverable clusters. Previous convex relaxation approaches achieving exact recovery do not allow any small clusters of size…
Decentralized optimization is well studied for smooth unconstrained problems. However, constrained problems or problems with composite terms are an open direction for research. We study structured (or composite) optimization problems, where…
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…
In this paper, we propose iterative inner/outer approximations based on a recent notion of block factor-width-two matrices for solving semidefinite programs (SDPs). Our inner/outer approximating algorithms generate a sequence of upper/lower…
A great variety of fundamental optimization and counting problems arising in computer science, mathematics and physics can be reduced to one of the following computational tasks involving polynomials and set systems: given an $m$-variate…
We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite…
For a symmetric positive semidefinite linear system of equations $\mathcal{Q} {\bf x} = {\bf b}$, where ${\bf x} = (x_1,\ldots,x_s)$ is partitioned into $s$ blocks, with $s \geq 2$, we show that each cycle of the classical block symmetric…
Mirror descent (MD) is a powerful first-order optimization technique that subsumes several optimization algorithms including gradient descent (GD). In this work, we develop a semi-definite programming (SDP) framework to analyze the…
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…
This paper introduces an abstract framework for randomized subspace correction methods for convex optimization, which unifies and generalizes a broad class of existing algorithms, including domain decomposition, multigrid, and block…
We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators,…
In many submodular optimization applications, datasets are naturally partitioned into disjoint subsets. These scenarios give rise to submodular optimization problems with partition-based constraints, where the desired solution set should be…
In recent years, copositive programming has received significant attention for its ability to model hard problems in both discrete and continuous optimization. Several relaxations of copositive programs based on semidefinite programming…
For general quadratically-constrained quadratic programming (QCQP), we propose a parabolic relaxation described with convex quadratic constraints. An interesting property of the parabolic relaxation is that the original non-convex feasible…
Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…
We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are…
The degree-$4$ Sum-of-Squares (SoS) SDP relaxation is a powerful algorithm that captures the best known polynomial time algorithms for a broad range of problems including MaxCut, Sparsest Cut, all MaxCSPs and tensor PCA. Despite being an…
Certificates of polynomial nonnegativity can be used to obtain tight dual bounds for polynomial optimization problems. We consider Sums of Nonnegative Circuit (SONC) polynomials certificates, which are well suited for sparse problems since…