knowing the fact that the main weakness of the most standard methods including k-means and hierarchical data clustering is their sensitivity to initialization and trapping to local minima, this paper proposes a modification of convex data clustering  in which there is no need to  be peculiar about how to select initial values. due to properly converting the task of optimization to an equivalent...

The ultimate goal of optimization is to find the minimizer of a target function. However, typical criteria for active optimization often ignore the uncertainty about the minimizer. We propose a novel criterion for global optimization and an associated sequential active learning strategy using Gaussian processes. Our criterion is the reduction of uncertainty in the posterior distribution of the ...

A normal quartic polynomial is a quartic polynomial whose fourth degree term coefficient tensor is positive definite. Its minimization problem is one of the simplest cases of nonconvex global optimization, and has engineering applications. We call a direction a global descent direction of a function at a point if there is another point with a lower function value along this direction. For a nor...

In this paper, a Þlled function for escaping the current local minimizer of a constrained global optimization problem is proposed. Then a Þlled function method for obtaining a global minimizer of the constrained optimization problem is presented. By this method, a global minimizer of a constrained global optimization problem can be obtained just by searching local minimizers of the original con...

S-quasiconvex functions (Phu and An, Optimization, Vol. 38, 1996) are stable with respect to the properties: “every lower level set is convex", “each local minimizer is a global minimizer", and “each stationary point is a global minimizer" (i.e., these properties remain true if a sufficiently small linear disturbance is added to a function of this class). In this paper, we introduce a subclass ...

We discuss two nonsmooth functions on Rn introduced by Nesterov. We show that the first variant is partly smooth in the sense of Lewis and that its only stationary point is the global minimizer. In contrast, we show that the second variant has 2n−1 Clarke stationary points, none of them local minimizers except the global minimizer, but also that its only Mordukhovich stationary point is the glo...

In this article we analyze the minimization of the so-called nonlocal isoperimetric problem (NLIP) posed on the flat 2-torus. After establishing regularity of the free boundary of minimizers, we show that when the parameter controlling the influence of the nonlocality is small, there is an interval of values for the mass constraint such that the global minimizer is exactly lamellar; that is, th...

We provide a simple counter-example to prove and illustrate that the backward differential flow approach, proposed by Zhu, Zhao and Liu for finding a global minimizer of coercive even-degree polynomials, can converge to a local minimizer rather than a global minimizer. We provide additional counter-examples to stress that convergence to a local minimum via the backward differential flow method ...

This paper considers the unconstrained global optimization with the revised filled function methods. The minimization sequence could leave from a local minimizer to a better minimizer of the objective function through minimizing an auxiliary function constructed at the local minimizer. Some promising numerical results are also included.

We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W , to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals ...