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Analyzing and reasoning about safety properties of software systems becomes an especially challenging task for programs with complex flow and, in particular, with loops or recursion. For such programs one needs additional information, for…

A critical problem in the emerging high-throughput genotyping protocols is to minimize the number of polymerase chain reaction (PCR) primers required to amplify the single nucleotide polymorphism loci of interest. In this paper we study PCR…

Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations…

This paper proposes QPALM, a proximal augmented Lagrangian method based on quadratic approximations, for solving nonlinear programming problems with weakly convex objective and constraint functions. The algorithm is constructed by…

We study a class of projective transformations of spectraplexes associated with self-dual cones and, on this basis, propose a polynomial-time algorithm for convex feasibility problems with positive definite constraints. At each iteration of…

We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to multiple polynomial inputs. For a given pair of polynomials and a degree, our algorithm…

#SMT, or model counting for logical theories, is a well-known hard problem that generalizes such tasks as counting the number of satisfying assignments to a Boolean formula and computing the volume of a polytope. In the realm of…

Loop invariants play a very important role in proving correctness of programs. In this paper, we address the problem of generating invariants of polynomial loop programs. We present a new approach, for generating polynomial equation…

In some fields such as Mathematics Mechanization, automated reasoning and Trustworthy Computing etc., exact results are needed. Symbolic computations are used to obtain the exact results. Symbolic computations are of high complexity. In…

We study nearly-linear time approximation algorithms for non-preemptive scheduling problems in two settings: the unrelated machine setting, and the identical machine with job precedence constraints setting, under the well-studied objectives…

We consider the problem of finding for a given $N$-tuple of polynomials (real or complex) the closest $N$-tuple that has a common divisor of degree at least $d$. Extended weighted Euclidean seminorm of the coefficients is used as a measure…

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…

Probabilistic numerical methods (PNMs) solve numerical problems via probabilistic inference. They have been developed for linear algebra, optimization, integration and differential equation simulation. PNMs naturally incorporate prior…

We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe…

We show that under mild assumptions for a problem whose solutions admit a dynamic programming-like recurrence relation, we can still find a solution under additional packing constraints, which need to be satisfied approximately. The number…

Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on…

Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we…

In this paper, we aim at introducing a new machine learning model, namely reconciled polynomial machine, which can provide a unified representation of existing shallow and deep machine learning models. Reconciled polynomial machine predicts…

We give a new fast method for evaluating sprectral approximations of nonlinear polynomial functionals. We prove that the new algorithm is convergent if the functions considered are smooth enough, under a general assumption on the spectral…

Primitive polynomials over finite fields are crucial for various domains of computer science, including classical pseudo-random number generation, coding theory and post-quantum cryptography. Nevertheless, the pursuit of an efficient…