Related papers: A class of nonlinear optimisation and applications
We characterize the class of probability measures on a compact Kahler manifold such that the associated Monge-Amp\`ere equation has a solution of finite pluricomplex energy. Our results are also valid in the big cohomology class setting.
An algorithm is proposed, analyzed, and tested for solving continuous nonlinear-equality-constrained optimization problems where the objective and constraint functions are defined by expectations or averages over large, finite numbers of…
We develop an $\e$-regularity theory at the boundary for a general class of Monge-Amp\`ere type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between H\"older densities supported on $C^2$…
We examine a multidimensional optimisation problem in the tropical mathematics setting. The problem involves the minimisation of a nonlinear function defined on a finite-dimensional semimodule over an idempotent semifield subject to linear…
We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This…
In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially…
We propose in this paper to study the solutions of some nonlinear elliptic equations with singular potential.
We first describe a general class of optimization problems that describe many natural, economic, and statistical phenomena. After noting the existence of a conserved quantity in a transformed coordinate system, we outline several instances…
In this chapter, I formulate four challenging conjectures in Nonlinear Analysis. More precisely: a conjecture on the Monge-Amp\`ere equation; a conjecture on an eigenvalue problem; a conjecture on a non-local problem; a conjecture on…
We study the problem of the existence and the holomorphicity of the Monge-Amp\`ere foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge-Amp\`ere equation even at points of arbitrary degeneracy. We obtain…
This paper proposes several novel optimization algorithms for minimizing a nonlinear objective function. The algorithms are enlightened by the optimal state trajectory of an optimal control problem closely related to the minimized objective…
We study the entropic regularization of the optimal transport problem in dimension 1 when the cost function is the distance c(x, y) = |y -- x|. The selected plan at the limit is, among those which are optimal for the non-penalized problem,…
We study the obstacle problem for a nonlocal, degenerate elliptic Monge--Amp\`ere equation. We show existence and regularity of a unique classical solution to the problem and regularity of the free boundary.
In this paper we consider a mass optimization problem in the case of scalar state function, where instead of imposing a constraint on the total mass of the competitors, we penalize the classical compliance by a convex functional defined on…
We study the optimal approximation of the solution of an operator equation Au=f by linear and nonlinear mappings. We identify those cases where optimal nonlinear approximation is better than optimal linear approximation.
A class of generalized nonlinear Kolmogorov equations is investigated. We present the group classification of Lie symmetries of the class with respect to the group of equivalence transformations. We find a number of exact solutions of…
A probabilistic method for solving the Monge-Kantorovich mass transport problem on $R^d$ is introduced. A system of empirical measures of independent particles is built in such a way that it obeys a doubly indexed large deviation principle…
Nonlinear programming problems are useful in designing and assigning work schedule and also in transporting goods and services from known sources to specified destinations. The objective function could be linear or nonlinear depending on…
In this paper, we consider a class of fully nonlinear equations on closed smooth Riemannian manifolds, which can be viewed as an extension of $\sigma_k$ Yamabe equation. Moreover, we prove local gradient and second derivative estimates for…
In this paper, we prove existence of $L^p$-optimal transport maps with $p \in (1,\infty)$ in a class of branching metric spaces defined on $\mathbb{R}^N$. In particular, we introduce the notion of cylinder-like convex function and we prove…