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In this work, we introduce the new class of functions which can use to solve the nonlinear/linear multi-dimensional differential equations. Based on these functions, a numerical method is provided which is called the Developed Lagrange…
The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using…
Many computer vision and human-computer interaction applications developed in recent years need evaluating complex and continuous mathematical functions as an essential step toward proper operation. However, rigorous evaluation of this kind…
This paper proposes a new algorithm for solving constrained global optimization problems where both the objective function and constraints are one-dimensional non-differentiable multiextremal Lipschitz functions. Multiextremal constraints…
In this paper, we study a class of fractional semi-infinite polynomial programming (FSIPP) problems, in which the objective is a fraction of a convex polynomial and a concave polynomial, and the constraints consist of infinitely many convex…
We introduce the following linear combination interpolation problem (LCI): Given $N$ distinct numbers $w_1,\ldots w_N$ and $N+1$ complex numbers $a_1,\ldots, a_N$ and $c$, find all functions $f(z)$ analytic in a simply connected set…
Hardware implementations of complex functions regularly deploy piecewise polynomial approximations. This work determines the complete design space of piecewise polynomial approximations meeting a given accuracy specification. Knowledge of…
This article presents an error analysis of the symmetric linear/bilinear partially penalized immersed finite element (PPIFE) methods for interface problems of Helmholtz equations. Under the assumption that the exact solution possesses a…
Basic principles of set theory have been applied in the context of probability and binary computation. Applying the same principles on inequalities is less common but can be extremely beneficial in a variety of fields. This paper formulates…
Distance functions between points in a domain are sometimes used to automatically plan a gradient-descent path towards a given target point in the domain, avoiding obstacles that may be present. A key requirement from such distance…
Additive utility function models are widely used in multiple criteria decision analysis. In such models, a numerical value is associated to each alternative involved in the decision problem. It is computed by aggregating the scores of the…
Nonlinear elliptic problems arise in many fields, including plasma physics, astrophysics, and optimal transport. In this article, we propose a novel operator-splitting/finite element method for solving such problems. We begin by introducing…
Many separable nonlinear optimization problems can be approximated by their nonlinear objective functions with piecewise linear functions. A natural question arising from applying this approach is how to break the interval of interest into…
In this paper we shall review the common problems associated with Piecewise Linear Separation incremental algorithms. This kind of neural models yield poor performances when dealing with some classification problems, due to the evolving…
We consider a popular family of constrained optimization problems arising in machine learning that involve optimizing a non-decomposable evaluation metric with a certain thresholded form, while constraining another metric of interest.…
The Performance Estimation Problem (PEP) approach consists in computing worst-case performance bounds on optimization algorithms by solving an optimization problem: one maximizes an error criterion over all initial conditions allowed and…
The eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a function basis that can be used in spectral analysis on manifolds. In [21] the authors recast the problem as an orthogonality constrained optimization problem…
In this work, we develop an adaptive, multivariate partitioning algorithm for solving mixed-integer nonlinear programs (MINLP) with multi-linear terms to global optimality. This iterative algorithm primarily exploits the advantages of…
The shortest path network interdiction (SPNI) problem poses significant computational challenges due to its NP-hardness. Current solutions, primarily based on integer programming methods, are inefficient for large-scale instances. In this…
Partitioned neural network functions are used to approximate the solution of partial differential equations. The problem domain is partitioned into non-overlapping subdomains and the partitioned neural network functions are defined on the…