The Stability Index Method (SIM) combines stochastic and deterministic algorithms to find global minima of multidimensional functions. The functions may be nonsmooth and may have multiple local minima. The method examines the change of the diameters of the minimizing sets for its stopping criterion. At first, the algorithm uses the uniform random distribution in the admissible set. Then normal ...

A novel approach for enclosing all heterogeneous and reactive azeotropes in multi-component mixtures is presented. The thermodynamic conditions for azeotropy form a system of nonlinear equations. A deterministic global optimization approach is introduced in which the global optimization problem may contain multiple global minima and there is a one-to-one correspondence between each global minim...

We study the problem of how to detect \interesting objects" appeared in a given image, I. Our approach is to treat it as a function approximation problem based on an over-redundant basis. Since the basis (a library of image templates) is over-redundant, there are innnitely many ways to decompose I. To select the \best" decomposition we rst propose a global optimization procedure that considers ...

A simulation algorithm is introduced, which uses a swarm of molecules to explore conformational space. The method uses multiple, different starting conformations and propagates them in time by integration of Newton’s equation of motion. In contrast to conventional molecular dynamics simulation of a set of independent molecules, in this method each molecule of the swarm is in addition subject to...

We study the problem of how to detect interesting objects appeared in a given image I Our approach is to treat it as a function approximation problem based on an over redundant basis Since the ba sis a library of image templates is over redundant there are in nitely many ways to decompose I To select the best decomposition we rst propose a global optimization procedure that considers a concave ...

Example 1. Imagine that we are solving a non-convex optimization problem on some (multivariate) function f using gradient descent. Recall that gradient descent converges to local minima. Because non-convex functions may have multiple minima, we cannot guarantee that gradient descent will converge to the global minimum. To resolve this issue, we will use random restarts, the process of starting ...

we consider the following problem min x2R n min kEk 2 k(A + E)x ? bk 2 where A is an m n real matrix and b is an n-dimensional real column vector, when it has multiple global minima. An eecient algorithm is presented to nd the global solution of minimum Euclidean norm.

Multiple layer solutions of the balanced bistable equation in innnite tubes are constructed via a variational method. I start with a characterization of Palais-Smale sequences which easily gives some global minima in the desired function classes as single layers. Assuming these minima are isolated as critical points, I paste them together to serve as an approximate multiple layer solution. If t...

We present a theory that enables us to construct heteroclinic connections in closed form for 2uxx = Wu(u), where x ∈ R, u(x) ∈ R and W is a smooth potential with multiple global minima. In particular, multiple connections between global minima are constructed for a class of potentials. With these potentials, numerical simulations for the vector AllenCahn equation ut = 22 ∆u−Wu(u) in two space d...

Multiple layer solutions of the balanced bistable equation in infinite tubes are constructed via a variational method. I start with a characterization of Palais-Smale sequences which easily gives some global minima in the desired function classes as single layers. Assuming these minima are isolated as critical points, I paste them together to serve as an approximate multiple layer solution. If ...