Related papers: Level set methods for finding critical points of m…
The problem of computing saddle points is important in certain problems in numerical partial differential equations and computational chemistry, and is often solved numerically by a minimization problem over a set of mountain passes. We…
We show that near any given minimizing sequence of paths for the mountain pass lemma, there exists a critical point whose polarization is also a critical point. This is motivated by the fact that if any polarization of a critical point is…
We show that computing even very coarse approximations of critical points is intractable for simple classes of nonconvex functions. More concretely, we prove that if there exists a polynomial-time algorithm that takes as input a polynomial…
This paper establishes the theoretical foundation for statistical applications of an intriguing new type of spatial point processes called critical point processes. These point processes, residing in Euclidean space, consist of the critical…
Level-set methods for convex optimization are predicated on the idea that certain problems can be parameterized so that their solutions can be recovered as the limiting process of a root-finding procedure. This idea emerges time and again…
We consider a kind of nonlinear systems on a locally finite graphs $G=(V,E)$. We prove via the mountain pass theorem that this kind of systems has a nontrivial ground state solution which depends on the parameter $\lambda$ with some…
The purpose of this paper is to establish a critical point theorem, which is an infinite-dimensional generalization of the classical generalized Mountain Pass Theorem of P. H. Rabinowitz \cite[Theorem 5.3]{Ra}. As application, we obtain the…
Computational difficulty of quadratic matching and the Gromov-Wasserstein distance has led to various approximation and relaxation schemes. One of such methods, relying on the notion of distance profiles, has been widely used in practice,…
We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a…
Convex optimization problems arising in applications often have favorable objective functions and complicated constraints, thereby precluding first-order methods from being immediately applicable. We describe an approach that exchanges the…
We study computing geometric problems on uncertain points. An uncertain point is a point that does not have a fixed location, but rather is described by a probability distribution. When these probability distributions are restricted to a…
Piecewise-linear (PL) Morse theory and discrete Morse theory are used in shape analysis tasks to investigate the topological features of discretized spaces. In spite of their common origin in smooth Morse theory, various notions of critical…
We study the Hermitian distance degree, a real enumerative invariant counting critical points of the squared Hermitian distance function, for matrix varieties invariant under left and right unitary actions. For such a variety \(M \subset…
In nearest-neighbor classification problems, a set of $d$-dimensional training points are given, each with a known classification, and are used to infer unknown classifications of other points by using the same classification as the nearest…
For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent $p$ for its $L^2$-normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations.…
The critical point is a fixed point in finite-size scaling. To quantify the behaviour of such a fixed point, we define, at a given temperature and scaling exponent ratio, the width of scaled observables for different sizes. The minimum of…
We study a general class of bilevel problems, consisting in the minimization of an upper-level objective which depends on the solution to a parametric fixed-point equation. Important instances arising in machine learning include…
In view of solving nonsmooth and nonconvex problems involving complex constraints (like standard NLP problems), we study general maximization-minimization procedures produced by families of strongly convex sub-problems. Using techniques…
We consider the problem of provably finding a stationary point of a smooth function to be minimized on the variety of bounded-rank matrices. This turns out to be unexpectedly delicate. We trace the difficulty back to a geometric obstacle:…
In this note we consider the classical variational tools like: Ekelenad's Variational Principle, Mountain Pass Lemma and some of their corollaries subject to a parameter. Next, we investigate the behaviour of critical points obtained once a…