Coupled polariton solitons in semiconductor microcavities with a double-well potential

In a double-well microcavity, coupling of spatial polariton solitons between the two wells is studied theoretically. Polaritons trapped inside the double-well structure can demonstrate bistability where both bright and dark polariton solitons are supported. Due to the tunneling effect, the combination of polariton solitons in the two wells can be bright-bright, dark-bright or dark-dark dependent on the initial states (i.e., on-state, off-state, or mixed-state) of the polaritons.     

On nonresonance impulsive functional differential inclusions with nonconvex valued right-hand side

In this paper we investigate the existence of solutions for first and second order nonresonance impulsive functional differential inclusions. We shall rely on the Schaefer's fixed point theorem combined with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values.     

On nonconvex optimization problems with separated nonconvex variables

A mathematical programming problem is said to have separated nonconvex variables when the variables can be divided into two groups: x=(x ₁,...,x _n ) and y=( y ₁,...,y _n ), such that the objective function and any constraint function is a sum of a convex function of (x, y) jointly and a nonconvex function of x alone. A method is proposed for solving a class of such problems which includes Lipschitz optimization, reverse convex programming problems and also more general nonconvex optimization problems.     

Ground waves in atomic chains with bi-monomial double-well potential

Ground waves in atomic chains are traveling waves that corresponds to minimal non-trivial critical values of the underlying action functional. In this paper we study FPU-type chains with bi-monomial double-well potential and prove the existence of both periodic and solitary ground waves. To this end we minimize the action on the Nehari manifold and show that periodic ground waves converge to solitary ones. Finally, we compute ground waves numerically by a suitable discretization of a constrained gradient flow.     

On constrained optimization with nonconvex regularization

Numerical Algorithms - In many engineering applications, it is necessary to minimize smooth functions plus penalty (or regularization) terms that violate smoothness and convexity. Specific...     

On nonconvex optimization with integral constraints

We consider the problem of minimizing f(y)dm with y dm=c,c fixed. The functionf is assumed to be continuous, but need not be convex. For this problem, we give necessary and sufficient conditions for the existence of solutions. We also give conditions under which uniqueness in a certain sense holds, and we show a relation which holds between the minimizers of two different problems and the corresponding values of the constraintsc.This research was supported by FINEP-Brazil, Grant Nos. 62.24-0416-00 and 4.2.82.0719-00.     

Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential

Abstract

We prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.     

On the extrema of band functions in periodic waveguides

In the work we construct an example of a periodic differential operator whose spectrum has gaps with edges attained by the band functions at interior points of the Brillouin zone. This example is the Laplacian on a pair of infinite parallel strips with common boundary from which a periodic system of small holes is cut out. At that, on the outer boundaries the Dirichlet condition is imposed, while on the common boundary the Neumann condition is considered.     

On optimal integration algorithms for functions with a known number of extrema

Translated from Programmnoe Obespechenie i Modeli Issledovaniya Operatsii, pp. 177–185, 1986.     

On a conjecture regarding the extrema of Bessel functions and its generalization

It was conjectured by Á. Elbert in J. Comput. Appl. Math. 133 (2001) 65-83 that, given two consecutive real zeros of a Bessel function of order ν, j_ν,κ and j_ν,κ+1, the zero of the derivative between such two zeros j_ν,κ′ satisfies . We prove that this inequality holds for any Bessel function of any real order. In addition to these lower bounds, upper bounds are obtained. In this way we bracket the zeros of the derivative. It is discussed how similar relations can be obtained for other special functions which are solutions of a second order ODE; in particular, the case of the zeros of is considered.     

On solving simple bilevel programs with a nonconvex lower level program

In this paper, we consider a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint and the upper level program has a convex set constraint. By using the value function of the lower level program, we reformulate the bilevel program as a single level optimization problem with a nonsmooth inequality constraint and a convex set constraint. To deal with such a nonsmooth and nonconvex optimization problem, we design a smoothing projected gradient algorithm for a general optimization problem with a nonsmooth inequality constraint and a convex set constraint. We show that, if the sequence of penalty parameters is bounded then any accumulation point is a stationary point of the nonsmooth optimization problem and, if the generated sequence is convergent and the extended Mangasarian-Fromovitz constraint qualification holds at the limit then the limit point is a stationary point of the nonsmooth optimization problem. We apply the smoothing projected gradient algorithm to the bilevel program if a calmness condition holds and to an approximate bilevel program otherwise. Preliminary numerical experiments show that the algorithm is efficient for solving the simple bilevel program.     

On the relative extrema of a linear combination of elementary symmetric functions

We determine where a linear combination of elementary symmetric functions attains as maximum and minimum over a certain convex set in Rⁿ . We also show that an inequality for elementary symmetric functions proposed by S. Pierce is true.     

On the relative extrema of a linear combination of elementary symmetric functions

We determine where a linear combination of elementary symmetric functions attains as maximum and minimum over a certain convex set in Rⁿ. We also show that an inequality for elementary symmetric functions proposed by S. Pierce is true.     

Splitting amplitudes of the lowest energy levels of the Schrödinger operator with double-well potential

An analytic method is proposed for calculating the asymptotic splitting of the lowest energy levels of the Schrödinger operator with a symmetric double-well potential. The potential describing a chain of pairwise interacting quantum particles in a common double-well potential is considered as an example. The limit of a large number of particles is investigated.Institute of Problems of Mechanics, Russian Academy of Sciences; Moscow Institute of Electronic Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 3, pp. 426–434, March, 1993.     

On a decomposition method for nonconvex global optimization

A rigorous foundation is presented for the decomposition method in nonconvex global optimization, including parametric optimization, partly convex, partly monotonic, and monotonic/linear optimization. Incidentally, some errors in the recent literature on this subject are pointed out and fixed.