Related papers: Formal Proofs for Nonlinear Optimization
We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy $Y^2=Y$, the matrix analog of binary variables that satisfy $z^2=z$, to model…
This paper considers robust solutions to a class of nonlinear least squares problems using min-max optimization approach. We give an explicit formula for the value function of the inner maximization problem and show the existence of global…
A number of recent results on optimization problems involving submodular functions have made use of the multilinear relaxation of the problem. These results hold typically in the value oracle model, where the objective function is…
In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order $s \in (0,1)$. We identify minimal conditions on the nonlinear term and the source which leads to existence of…
Cutting plane methods, particularly outer approximation, are a well-established approach for solving nonlinear discrete optimization problems without relaxing the integrality of decision variables. While powerful in theory, their…
We develop a framework which allows us to prove the essential general quasi-orthogonality for the non-symmetric Johnson-Nedelec finite element/boundary element coupling. General quasi-orthogonality was first proposed in [Axioms of…
We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a nondecreasing submodular…
Nonnegativity certificates can be used to obtain tight dual bounds for polynomial optimization problems. Hierarchies of certificate-based relaxations ensure convergence to the global optimum, but higher levels of such hierarchies can become…
Topology optimization of frame structures under free-vibration eigenvalue constraints constitutes a challenging nonconvex polynomial optimization problem with disconnected feasible sets. In this article, we first formulate it as a…
In this paper we present a unified framework for solving a general class of problems arising in the context of set-membership estimation/identification theory. More precisely, the paper aims at providing an original approach for the…
Weighted Max-SAT is the optimization version of SAT and many important problems can be naturally encoded as such. Solving weighted Max-SAT is an important problem from both a theoretical and a practical point of view. In recent years, there…
For the problem of maximizing a monotone, submodular function with respect to a cardinality constraint $k$ on a ground set of size $n$, we provide an algorithm that achieves the state-of-the-art in both its empirical performance and its…
Meshing of geometric domains having curved boundaries by affine simplices produces a polytopial approximation of those domains. The resulting error in the representation of the domain limits the accuracy of finite element methods based on…
This work presents a unified framework that combines global approximations with locally built models to handle challenging nonconvex and nonsmooth composite optimization problems, including cases involving extended real-valued functions. We…
In this paper, we concentrate on a particular category of quadratically constrained quadratic programming (QCQP): nonconvex QCQP with one equality constraint. This type of QCQP problem optimizes a quadratic objective under a fixed…
With the extensive application of submodularity, its generalizations are constantly being proposed. However, most of them are tailored for special problems. In this paper, we focus on quasi-submodularity, a universal generalization, which…
Solving non-linear Diophantine systems lies at the mathematical core of integer optimization and cryptography. While the general unbounded problem is undecidable, even over bounded integer domains it remains classically intractable in the…
This paper considers the minimization of a continuously differentiable function over a cardinality constraint. We focus on smooth and relatively smooth functions. These smoothness criteria result in new descent lemmas. Based on the new…
We present an optimization algorithm that can identify a global minimum of a potentially nonconvex smooth function with high probability, assuming the Gibbs measure of the potential satisfies a logarithmic Sobolev inequality. Our…
In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several…