Lagrange-type duality in DC programming

We show a Lagrange-type duality theorem for a DC programming problem, which is a generalization of previous results by J.-E. Martínez-Legaz, M. Volle [5] and Y. Fujiwara, D. Kuroiwa [1] when all constraint functions are real-valued. To the purpose, we decompose the DC programming problem into certain infinite convex programming problems.     

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

In this paper, we prove the existence of 12 small-amplitude limit cycles around a singular point in a planar cubic-degree polynomial system. Based on two previously developed cubic systems in the literature, which have been proved to exhibit 11 small-amplitude limit cycles, we applied a different method to show 11 limit cycles. Moreover, we show that one of the systems can actually have 12 small-amplitude limit cycles around a singular point. This is the best result so far obtained in cubic planar vector fields around a singular point.     

Performance analysis of linear and nonlinear techniques for automatic scaling of discretized control problems

This paper introduces three (one linear and two nonlinear) automatic scaling techniques for NLPs with states and constraints spread over several orders of magnitude, without requiring complex off-the-shelf external tools. All of these methods have been compared to standard techniques and applied to three problems using SNOPT and IPOPT. The results confirm that the proposed techniques significantly improve the NLP conditioning, yielding more reliable and in some cases, faster NLP solutions.     

Approximating Pareto curves using semidefinite relaxations

We approximate as closely as desired the Pareto curve associated with bicriteria polynomial optimization problems. We use three formulations (including the weighted sum approach and the Chebyshev approximation) and each of them is viewed as a parametric polynomial optimization problem. For each case is associated a hierarchy of semidefinite relaxations and from an optimal solution of each relaxation one approximates the Pareto curve by solving an inverse problem (first two cases) or by building a polynomial underestimator (third case).     

A Remark on the Cauchy Problem of 1D Compressible Navier-Stokes Equations with Density-dependent Viscosity Coefcients

This paper is concerned with the Cauchy problems of one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefcients.By assumingρ0∈L1(R),we will prove the existence of weak solutions to the Cauchy problems forθ0.This will improve results in Jiu and Xin’s paper(Kinet.Relat.Models,1(2):313–330(2008))in whichθ12is required.In addition,We will study the large time asymptotic behavior of such weak solutions.     

Confidence-based optimisation for the newsvendor problem under binomial,Poisson and exponential demand

We introduce a novel strategy to address the issue of demand estimation in single-item single-period stochastic inventory optimisation problems. Our strategy analytically combines confidence interval analysis and inventory optimisation. We assume that the decision maker is given a set of past demand samples and we employ confidence interval analysis in order to identify a range of candidate order quantities that, with prescribed confidence probability, includes the real optimal order quantity for the underlying stochastic demand process with unknown stationary parameter(s). In addition, for each candidate order quantity that is identified, our approach produces an upper and a lower bound for the associated cost. We apply this approach to three demand distributions in the exponential family: binomial, Poisson, and exponential. For two of these distributions we also discuss the extension to the case of unobserved lost sales. Numerical examples are presented in which we show how our approach complements existing frequentist—e.g. based on maximum likelihood estimators—or Bayesian strategies.     

A semilocal convergence result for Newton’s method under generalized conditions of Kantorovich

From Kantorovich’s theory we establish a general semilocal convergence result for Newton’s method based fundamentally on a generalization required to the second derivative of the operator involved. As a consequence, we obtain a modification of the domain of starting points for Newton’s method and improve the a priori error estimates. Finally, we illustrate our study with an application to a special case of conservative problems.     

吉普-加油站问题

利用最优序列方法研究了吉普－加油站问题,确定了单向行驶吉普－加油站问题和往返行驶吉普－加油站问题的最优序列。