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We present a finite-horizon optimization algorithm that extends the established concept of Dual Dynamic Programming (DDP) in two ways. First, in contrast to the linear costs, dynamics, and constraints of standard DDP, we consider problems…

The Lasserre/Sum-of-Squares (SoS) hierarchy is a systematic procedure for constructing a sequence of increasingly tight semidefinite relaxations. It is known that the hierarchy converges to the 0/1 polytope in n levels and captures the…

Stochastic Barrier Functions (SBFs) certify the safety of stochastic systems by formulating a functional optimization problem, which state-of-the-art methods solve using Sum-of-Squares (SoS) polynomials. This work focuses on polynomial SBFs…

Symbolic Mathematical tasks such as integration often require multiple well-defined steps and understanding of sub-tasks to reach a solution. To understand Transformers' abilities in such tasks in a fine-grained manner, we deviate from…

This work presents a computationally efficient approach to data-driven robust contracting controller synthesis for polynomial control-affine systems based on a sum-of-squares program. In particular, we consider the case in which a system…

We study SOS properties of biquadratic forms. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic…

We show that if a system of degree-$k$ polynomial constraints on~$n$ Boolean variables has a Sums-of-Squares (SOS) proof of unsatisfiability with at most~$s$ many monomials, then it also has one whose degree is of the order of the square…

In polynomial optimization problems, nonnegativity constraints are typically handled using the sum of squares condition. This can be efficiently enforced using semidefinite programming formulations, or as more recently proposed by Papp and…

For Open Source Software (OSS) projects, discussions in Issue Tracking Systems (ITS) serve as a crucial collaboration mechanism for diverse stakeholders. However, these discussions can become lengthy and entangled, making it hard to find…

With a high probability the Sarlos randomized algorithm of 2006 outputs a nearly optimal least squares solution of a highly overdeterminedlinear system of equations. We propose its simple deterministic variation which computes such a…

We describe an approximate dynamic programming method for stochastic control problems on infinite state and input spaces. The optimal value function is approximated by a linear combination of basis functions with coefficients as decision…

The Sum-of-Squares (SoS) hierarchy, also known as Lasserre hierarchy, has emerged as a promising tool in optimization. However, it remains unclear whether fixed-degree SoS proofs can be automated [O'Donnell (2017)]. Indeed, there are…

We exhibit families of $4$-CNF formulas over $n$ variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) $d$ but require SOS proofs of size $n^{\Omega(d)}$ for values of $d = d(n)$ from constant all the…

There is an increasing demand for scalable algorithms capable of clustering and analyzing large time series datasets. The Kohonen self-organizing map (SOM) is a type of unsupervised artificial neural network for visualizing and clustering…

The Boolean Satisfiability problem (SAT), as the prototypical $\mathsf{NP}$-complete problem, is crucial in both theoretical computer science and practical applications. To address this problem, stochastic local search (SLS) algorithms,…

Given $(a_1, \dots, a_n, t) \in \mathbb{Z}_{\geq 0}^{n + 1}$, the Subset Sum problem ($\mathsf{SSUM}$) is to decide whether there exists $S \subseteq [n]$ such that $\sum_{i \in S} a_i = t$. There is a close variant of the $\mathsf{SSUM}$,…

In this paper, we propose a general sparse decomposition of dynamical systems provided that the vector field and constraint set possess certain sparse structures, which we call subsystems. This notion is based on causal dependence in the…

SOL is an open-source library for scalable online learning algorithms, and is particularly suitable for learning with high-dimensional data. The library provides a family of regular and sparse online learning algorithms for large-scale…

The SCIP Optimization Suite provides a collection of software packages for mathematical optimization, centered around the constraint integer programming (CIP) framework SCIP. This report discusses the enhancements and extensions included in…

In a common formulation of semi-infinite programs, the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval. If this function is sufficiently closely approximable by…