Related papers: SoS certification for symmetric quadratic function…
Decomposition of any Boolean Function BF_n of n binary inputs into an optimal inverter coupled network of Symmetric Boolean functions SF_k (k \leq n) is described. Each SF component is implemented by Threshold Logic Cells, forming a…
The Sum-of-Squares (SOS) approximation method is a technique used in optimization problems to derive lower bounds on the optimal value of an objective function. By representing the objective function as a sum of squares in a feature space,…
The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit…
Global optimization has gained attraction over the past decades, thanks to the development of both theoretical foundations and efficient numerical routines. Among recent advances, Kernel Sum of Squares (KernelSOS) provides a powerful…
Certifying nonnegativity of polynomials is a well-known NP-hard problem with direct applications spanning non-convex optimization, control, robotics, and beyond. A sufficient condition for nonnegativity is the Sum of Squares (SOS) property,…
The sum-of-squares (SoS) hierarchy is a powerful technique based on semi-definite programming that can be used for both classical and quantum optimization problems. This hierarchy goes under several names; in particular, in quantum…
We address black-box convex optimization problems, where the objective and constraint functions are not explicitly known but can be sampled within the feasible set. The challenge is thus to generate a sequence of feasible points converging…
This paper is devoted to the problem of minimizing a sum of rational functions over a basic semialgebraic set. We provide a hierarchy of sum of squares (SOS) relaxations that is dual to the generalized moment problem approach due to…
Consider a system of $m$ polynomial equations $\{p_i(x) = b_i\}_{i \leq m}$ of degree $D\geq 2$ in $n$-dimensional variable $x \in \mathbb{R}^n$ such that each coefficient of every $p_i$ and $b_i$s are chosen at random and independently…
We give an algorithm for solving unique games (UG) instances whenever low-degree sum-of-squares proofs certify good bounds on the small-set-expansion of the underlying constraint graph via a hypercontractive inequality. Our algorithm is in…
We consider the problem of approximating Quadratic O-1 Integer Programs with bounded number of constraints and non-negative constraint matrix entries, which we term as PIQP. We describe and analyze a randomized algorithm based on a program…
This paper addresses the optimization problem of minimizing non-convex continuous functions, which is relevant in the context of high-dimensional machine learning applications characterized by over-parametrization. We analyze a randomized…
We introduce an S.o.S hierarchy of lower bounds for a polynomial optimization problem whose constraint is expressed as a matrix polynomial semidefinite inequality. Our approach involves utilizing a penalty function framework to directly…
The sum-of-squares hierarchy of semidefinite programs has become a common tool for algorithm design in theoretical computer science, including problems in quantum information. In this work we study a connection between a Hermitian version…
We investigate the maximum sum-of-squares (SOS) rank of biquadratic forms in the critical case of $4 \times 3$ variables, where the general bounds are currently $7 \leq \mathrm{BSR}(4,3) \leq 11$. By analyzing two important structured…
The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, control, machine learning, game theory, and combinatorics, among others.…
In this paper, we propose a new convergent conic programming hierarchy of relaxations involving both semi-definite cone and second-order cone constraints for solving nonconvex polynomial optimization problems to global optimality. The…
We analyze the sum-of-squares rank of unweighted instances of the Minimum Knapsack (MK) problem, i.e., minimization of $\sum_{i=1}^n x_i$ for 0/1 variables under the constraint $\sum_{i=1}^n x_i \geq q$, with $q \in \mathbb{R}$. Such…
A line of work initiated by Fortnow in 1997 has proven model-independent time-space lower bounds for the $\mathsf{SAT}$ problem and related problems within the polynomial-time hierarchy. For example, for the $\mathsf{SAT}$ problem, the…
Many previous Sum-of-Squares (SOS) lower bounds for CSPs had two deficiencies related to global constraints. First, they were not able to support a "cardinality constraint", as in, say, the Min-Bisection problem. Second, while the…