In this paper we present a Lagrange-multiplier formulation of discrete constrained optimization problems, the associated discrete-space first-order necessary and sufficient conditions for saddle points, and an efficient first-order search procedure that looks for saddle points in discrete space. Our new theory provides a strong mathematical foundation for solving general nonlinear discrete opti...

Bifurcation theory provides a classification of the expected ways in which the number and/or stability of invariant solutions (‘attractors’ and ‘repellors’) of nonlinear ordinary differential equations may change as parameter values are changed. The most common qualitative changes are ‘saddle-node’ bifurcations, ‘Hopf’ bifurcations, and ‘SNIPER’ bifurcations. At a saddle-node bifurcation, a pai...

In case f0 and C are convex, a variety of primal/dual numerical methods can be used to find the saddle points of a Lagrangian L associated with this problem [4, 7]. These methods take advantage of the fact that the search for the saddle points of L is unconstrained, or conducted over sets much simpler than C. Furthermore, we can introduce penalties on f0 in a way that regularizes L and makes it...

We discuss O(N) invariant scalar field theories in 0+1 and 1+1 space-time dimensions. Combining ordinary “Large N” saddle point techniques and simple properties of the diagonal resolvent of one dimensional Schrödinger operators we find exact non-trivial (space dependent) solutions to the saddle point equations of these models in addition to the saddle point describing the ground state of the th...

Finding multiple local minima in the merit function landscape of optical system optimization is a difficult task, especially for complex designs that have a large number of variables. We discuss here a method that enables a rapid generation of new local minima for optical systems of arbitrary complexity. We have recently shown that saddle points known in mathematics as Morse index 1 saddle poin...

We present a theory of rare events and derive an algorithm to obtain rates from postprocessing the numerical data of a free energy calculation and the corresponding committor analysis. The formalism is based on the division of the saddle region of the free energy profile of the rare event into two adjacent segments called saddle domains. The method is built on sampling the dynamics within these...

Chaotic saddles are unstable invariant sets in the phase space of dynamical systems that exhibit transient chaos. They play a key role in mediating transport processes involving scattering and chaotic transients. Here we present evidence (long chaotic transients and fractal basins of attraction) of transient chaos in a "logistic" delay differential equation. We adapt an existing method (stagger...