New exact penalty function for solving constrainedfinite min-max problems

This paper introduces a new exact and smooth penalty function to tackle constrained min-max problems. By using this new penalty function and adding just one extra variable, a constrained rain-max problem is transformed into an unconstrained optimization one. It is proved that, under certain reasonable assumptions and when the penalty parameter is sufficiently large, the minimizer of this unconstrained optimization problem is equivalent to the minimizer of the original constrained one. Numerical results demonstrate that this penalty function method is an effective and promising approach for solving constrained finite min-max problems.

An entropy based central cutting plane algorithm for convex min-max semi-infinite programming problems

In this paper,we present a central cutting plane algorithm for solving convex min-max semi-infinite programming problems.Because the objective function here is non-differentiable,we apply a smoothing technique to the considered problem and develop an algorithm based on the entropy function.It is shown that the global convergence of the proposed algorithm can be obtained under weaker conditions.Some numerical results are presented to show the potential of the proposed algorithm.

Optimal paths planning in dynamic transportation networks with random link travel times

A theoretical study was conducted on finding optimal paths in transportation networks where link travel times were stochastic and time-dependent(STD). The methodology of relative robust optimization was applied as measures for comparing time-varying, random path travel times for a priori optimization. In accordance with the situation in real world, a stochastic consistent condition was provided for the STD networks and under this condition, a mathematical proof was given that the STD robust optimal path problem can be simplified into a minimum problem in specific time-dependent networks. A label setting algorithm was designed and tested to find travelers' robust optimal path in a sampled STD network with computation complexity of O(n2+n·m). The validity of the robust approach and the designed algorithm were confirmed in the computational tests. Compared with conventional probability approach, the proposed approach is simple and efficient, and also has a good application prospect in navigation system.

Controlling interstory drift ratio profiles via topology optimization strategies

An approach to control the profiles of interstory drift ratios along the height of building structures via topology optimization is proposed herein.The theoretical foundation of the proposed approach involves solving a min-max optimization problem to suppress the maximum interstory drift ratio among all stories.Two formulations are suggested:one inherits the bound formulation and the other utilizes a p-norm function to aggregate all individual interstory drift ratios.The proposed methodology can shape the interstory drift ratio profiles into inverted triangular or quadratic patterns because it realizes profile control using a group of shape weight coefficients.The proposed formulations are validated via a series of numerical examples.The disparity between the two formulations is clear.The optimization results show the optimal structural features for controlling the interstory drift ratios under different requirements.

Option Pricing when the Regime-Switching Risk is Priced

We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset are modulated by an observable continuous-time, finite-state Markov chain. We develop a two- stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the option prices is significant.

Optimizing the product of nonzero functions

OPTIMIZING problems on the product of a number of functions are generally quite complex,even if each factor function is very simple. In this note, a simplified method is established foroptimizing the product of non-zero functions. It is proved that the problem is equivalent to opti-

Robust hypothesis testing for asymmetric nominal densities under a relative entropy tolerance

In this paper, we address an open problem raised by Levy(2009) regarding the design of a binary minimax test without the symmetry assumption on the nominal conditional probability densities of observations. In the binary minimax test, the nominal likelihood ratio is a monotonically increasing function and the probability densities of the observations are located in neighborhoods characterized by placing a bound on the relative entropy between the actual and nominal densities. The general minimax testing problem at hand is an infinite-dimensional optimization problem, which is quite difficult to solve. In this paper, we prove that the complicated minimax testing problem can be substantially reduced to solve a nonlinear system of two equations having only two unknown variables, which provides an efficient numerical solution.

Quasi-Optimized Overlapping Schwarz Waveform Relaxation Algorithm for PDEs with Time-Delay

Schwarzwaveformrelaxation(SWR)algorithmhas been investigated deeply and widely for regular time dependent problems.But for time delay problems,complete analysis of the algorithm is rare.In this paper,by using the reaction diffusion equations with a constant discrete delay as the underlying model problem,we investigate the convergence behavior of the overlapping SWR algorithm with Robin transmission condition.The key point of using this transmission condition is to determine a free parameter as better as possible and it is shown that the best choice of the parameter is determined by the solution of a min-max problem,which is more complex than the one arising for regular problems without delay.We propose new notion to solve the min-max problem and obtain a quasi-optimized choice of the parameter,which is shown efficient to accelerate the convergence of the SWR algorithm.Numerical results are provided to validate the theoretical conclusions.