Research Article
Racing Sampling Based Microimmune Optimization Approach
Solving Constrained Expected Value Programming
Kai Yang
1
and Zhuhong Zhang
2
1
College of Computer Science, Guizhou University, Guiyang 550025, China
2
Department of Big Data Science and Engineering, College of Big Data and Information Engineering, Guizhou University,
Guiyang 550025, China
Correspondence should be addressed to Zhuhong Zhang; sci.zhzhang@gzu.edu.cn
Received 24 December 2015; Accepted 23 February 2016
Academic Editor: Eduardo Rodr
´
ıguez-Tello
Copyright © 2016 K. Yang and Z. Zhang. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
is work investigates a bioinspired microimmune optimization algorithm to solve a general kind of single-objective nonlinear
constrained expected value programming without any prior distribution. In the study of algorithm, two lower bound sample
estimates of random variables are theoretically developed to estimate the empirical values of individuals. Two adaptive racing
sampling schemes are designed to identify those competitive individuals in a given population, by which high-quality individuals
can obtain large sampling size. An immune evolutionary mechanism, along with a local search approach, is constructed to evolve
the current population. e comparative experiments have showed that the proposed algorithm can eectively solve higher-
dimensional benchmark problems and is of potential for further applications.
1. Introduction
Many real-world engineering optimization problems, such
as industrial control, project management, portfolio invest-
ment, and transportation logistics, include stochastic param-
eters or random variables usually. Generally, they can be
solved by some existing intelligent optimization approaches
with static sampling strategies (i.e., each candidate is with
the same sampling size), aer being transformed into con-
strained expected value programming (CEVP), chance con-
strained programming, or probabilistic optimization models.
Although CEVP is a relatively simple topic in the context
of stochastic programming, it is a still challenging topic,
as it is dicult to nd feasible solutions and meanwhile
the quality of the solution depends greatly on environ-
mental disturbance. e main concern of solving CEVP
involves two aspects: (i) when stochastic probabilistic dis-
tributions are unknown, it becomes crucial to distinguish
those high-quality individuals from the current population
in uncertain environments, and (ii) although static sam-
pling strategies are a usual way to handle random factors,
the expensive computational cost is inevitable, and hence
adaptive sampling strategies with low computational cost are
desired.
When stochastic characteristics are unknown, CEVP
models are usually replaced by their sample average approx-
imation models [1, 2], and thereaer some new or existing
techniques can be used to nd their approximate solutions.
Mathematically, several researchers [3–5] probed into the
relationshipbetweenCEVPmodelsandtheirapproximation
onesandacquiredsomevaluablelowerboundestimates
on sample size capable of being used to design adaptive
sampling rules. On the other hand, intelligent optimiza-
tion techniques have become popular for nonconstrained
expected value programming problems [6–8], in which
some advanced sampling techniques, for example, adaptive
sampling techniques and sample allocation schemes, can
eectively suppress environmental inuence on the process
of solution search. Unfortunately, studies on general CEVP
have been rarely reported in the literature because of expected
value constraints. Even if so, several researchers made great
eorts to investigate new or hybrid intelligent optimization
approaches for such kind of uncertain programming prob-
lem. For example, B. Liu and Y.-K. Liu [9] proposed a hybrid
Hindawi Publishing Corporation
Scientific Programming
Volume 2016, Article ID 2148362, 9 pages
http://dx.doi.org/10.1155/2016/2148362