Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points such that the first and second order derivatives cannot distinguish them with l...

We present a non-smooth optimization technique for non-convex maximum eigenvalue functions and for non-smooth functions which are infinite maxima of eigenvalue functions. We prove global convergence of our method in the sense that for an arbitrary starting point, every accumulation point of the sequence of iterates is critical. The method is tested on several problems in feedback control synthe...

1.1. Gaussian random fields on manifolds. Let X(p), p ∈ M̃ , be a realvalued random field on an orientable manifold M̃ . In this article, we consider a Gaussian random field with a smooth sample path and one-dimensional standard normal marginals (i.e., X(p) ∼ N(0, 1) for each p ∈ M̃). Let r(p, q) = Cov(X(p), X(q)) denote the covariance function. If r(p, q) is sufficiently smooth in a neighborhood ...

A local convergence result for abstract descent methods is proved. The sequence of iterates is attracted by a local (or global) minimum, stays in its neighborhood and converges. This result allows algorithms to exploit local properties of the objective function: The gradient of the Moreau envelope of a prox-regular functions is locally Lipschitz continuous and expressible in terms of the proxim...

We present TangentBug, a new ran.ge-sensor based navigation algorithm for two degrees-o,F-freedom mobile robots. The algorithm combines local reactive planning with globally convergent behavior. For the local planning, TangentBug uses the range d a t a to compute a locally shortest path based on a novel structure termed the local tangent graph, or LTG. The robot uses the LTG for choosing the lo...

We consider the problem of optimal partitional clustering of real data sets by optimizing three basic criteria (trace of within scatter matrix, variance ratio criterion, and Marriottt’s criterion). Four variants of the algorithm based on differential evolution with competing strategies are compared on eight real-world data sets. The experimental results showed that hybrid variants with k-means ...

A cardinality constraint imposes that each value of a set V must be taken a certain number of times by a set of variables X, whereas an among constraint imposes that a certain number of variables of a set X must take a value in the set V. This paper studies several combinations of among constraints and several conjunctions of among constraints and cardinality constraints. Some filtering algorit...

We study the dynamics of a social network. Each node has to decide locally which other node it wants to befriend, i.e., to which other node it wants to create a connection in order to maximize its welfare, which is defined as the sum of the weights of incident edges. This allows us to model the cooperation between nodes where every node tries to do as well as possible. With the limitation that ...

Consider the multi-homogeneous homotopy continuation method for solving a system of polynomial equations. For any partition of variables, the multi-homogeneous Bézout number bounds the number of isolated solution curves one has to follow in the method. This paper presents a local search method for finding a partition of variables with minimal multi-homogeneous Bézout number. As with any other l...

In this paper, we propose a stochastic dynamic local search (SDLS) method for Multiple-Valued Logic (MVL) learning by introducing stochastic dynamics into the traditional local search method. The proposed learning network maintains some trends of quick descent to either global minimum or a local minimum, and at the same time has some chance of escaping from local minima by permitting temporary ...