Related papers: A non-linear monotonicity principle and applicatio…
We revisit the following nonlinear Schr\"odinger system \begin{align*}\begin{cases} -\epsilon^{2}\Delta u +P(x) u= \mu_1 u^3 +\beta uv^2, &~\text{in}\;\mathbb {R}^3,\\ -\epsilon^{2}\Delta v+Q(x) v= \mu_2 v^3 +\beta u^2v,…
We introduce graphs associated to transport problems between discrete marginals, that allow to characterize the set of all optimizers given one primal optimizer. In particular, we establish that connectivity of those graphs is a necessary…
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 develop an elementary and self-contained differential approach, in an $L^\infty$ setting, for well-posedness (existence, uniqueness and smooth dependence with respect to the data) for the multi-marginal Schr{\"o}dinger system which…
A counterexample to uniqueness of global minimizers of semilinear optimal control problems is given. The lack of uniqueness occurs for a special choice of the state-target in the cost functional. Our arguments show also that, for some…
This paper is concerned with six variational problems and their mutual connections: The quadratic Monge-Kantorovich optimal transport, the Schr\"odinger problem, Brenier's relaxed model for incompressible fluids, the so-called Br\"odinger…
The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian (pre-)length spaces, including cyclical monotonicity, stability of optimal couplings and Kantorovich duality…
We study global optimization of non-convex functions through optimal control theory. Our main result establishes that (quasi-)optimal trajectories of a discounted control problem converge globally and practically asymptotically to the set…
Replacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\"odinger. This approach…
We consider optimal route planning when the objective function is a general nonlinear and non-monotonic function. Such an objective models user behavior more accurately, for example, when a user is risk-averse, or the utility function needs…
This paper describes new results linking constrained optimization theory and nonlinear contraction analysis. Generalizations of Lagrange parameters are derived based on projecting system dynamics on the tangent space of possibly…
Real-life problems are governed by equations which are nonlinear in nature. Nonlinear equations occur in modeling problems, such as minimizing costs in industries and minimizing risks in businesses. A technique which does not involve the…
In this paper, we introduce a class of nonlinear optimisation problems. Under mild assumptions, we obtain the existence of potential functions and show that the potential function is a generalised solution of a Monge-Amp\`ere type equation.…
Leveraging connections between diffusion-based sampling, optimal transport, and stochastic optimal control through their shared links to the Schr\"odinger bridge problem, we propose novel objective functions that can be used to transport…
I study heat and norm transport in a one-dimensional lattice of linear Schr\"odinger oscillators with conservative stochastic perturbations. Its equilibrium properties are the same of the Discrete Nonlinear Schr\"odinger equation in the…
This work is a geometrical approach to the optimization problem motivated by transportation system management. First, an attempt has been made to furnish a comprehensive account of geometric programming based on the elementary Finsler…
It is demonstrated that nonlinear dynamical systems with analytic nonlinearities can be brought down to the abstract Schr\"odinger equation in Hilbert space with boson Hamiltonian. The Fourier coefficients of the expansion of solutions to…
We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying system. Our strategy combines the construction of quasi-turnpike controls via…
This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e. the original problem is first reformulated as a nonconvex optimization problem, its well-posedness…
Finding optimal trajectories for multiple traffic demands in a congested network is a challenging task. Optimal transport theory is a principled approach that has been used successfully to study various transportation problems. Its usage is…