Related papers: Multi-Degrees in Polynomial Optimization
Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation, and model selection. These are all optimization problems, well-known to be challenging due to…
In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems considering subsets of the objective functions. If the…
Global optimization of decision trees is a long-standing challenge in combinatorial optimization, yet such models play an important role in interpretable machine learning. Although the problem has been investigated for several decades, only…
We provide a closed formula for the degree of $\text{SO}(n)$ over an algebraically closed field of characteristic zero. In addition, we describe symbolic and numerical techniques which can also be used to compute the degree of…
Gradient-based methods are widely used to solve various optimization problems, however, they are either constrained by local optima dilemmas, simple convex constraints, and continuous differentiability requirements, or limited to…
The paper aims to investigate relevant computational issues of deep neural network architectures with an eye to the interaction between the optimization algorithm and the classification performance. In particular, we aim to analyze the…
We consider the problem of finding a subgraph of a given graph which maximizes a given function evaluated at its degree sequence. While the problem is intractable already for convex functions, we show that it can be solved in polynomial…
This article focuses on the optimization of a complex system which is composed of several subsystems. On the one hand, these subsystems are subject to multiple objectives, local constraints as well as local variables, and they are…
This note will study complex polynomial maps of degree $n\ge 2$ with only one critical point.
This is a survey on the use of low-degree polynomials to predict and explain the apparent statistical-computational tradeoffs in a variety of average-case computational problems. In a nutshell, this framework measures the complexity of a…
We consider the problem of finding the best memoryless stochastic policy for an infinite-horizon partially observable Markov decision process (POMDP) with finite state and action spaces with respect to either the discounted or mean reward…
In this short note, we discuss a goal-oriented multiobjective optimization problem for system performance assessment. The objective function for such optimization problem, which is usually a composite of different performance indices…
While several classes of integer linear optimization problems are known to be solvable in polynomial time, far fewer tractability results exist for integer nonlinear optimization. In this work, we narrow this gap by identifying a broad…
We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling…
In this article we combine two developments in polynomial optimization. On the one hand, we consider nonnegativity certificates based on sums of nonnegative circuit polynomials, which were recently introduced by the second and the third…
This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints…
In this paper we propose a method that uses Lagrange multipliers and numerical algebraic geometry to find all critical points, and therefore globally solve, polynomial optimization problems. We design a polyhedral homotopy algorithm that…
We present a geometric multilevel optimization approach that smoothly incorporates box constraints. Given a box constrained optimization problem, we consider a hierarchy of models with varying discretization levels. Finer models are…
Necessary conditions for high-order optimality in smooth nonlinear constrained optimization are explored and their inherent intricacy discussed. A two-phase minimization algorithm is proposed which can achieve approximate first-, second-…
This paper investigates a category of constrained fractional optimization problems that emerge in various practical applications. The objective function for this category is characterized by the ratio of a numerator and denominator, both…