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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 965835, 13 pages
Tractable Approximation to Robust Nonlinear Production Frontier Problem
1Department of Economic Mathematics, South Western University of Finance and Economics, Sichuan, Chengdu 610074, China
2Department of Mathematics, Sichuan University, Chengdu 610064, China
Received 4 September 2012; Accepted 24 September 2012
Academic Editor: Jen-Chih Yao
Copyright © 2012 Lei Wang et al. This 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.
Robust optimization is a rapidly developing methodology for handling optimization problems affected by the uncertain-but-bounded data perturbations. In this paper, we consider the nonlinear production frontier problem where the traditional expected linear cost minimization objective is replaced by one that explicitly addresses cost variability. We propose a robust counterpart for the nonlinear production frontier problem that preserves the computational tractability of the nominal problem. We also provide a guarantee on the probability that the robust solution is feasible when the uncertain coefficients obey independent and identically distributed normal distributions.
Optimization is a leading methodology in engineering design and control. Recently the robust optimization methodology has been introduced and studied [1–10] in order to deal with uncertain optimization problems: those for which part or all of their parameters are uncertain or inexact, or those for which the computed optimal solution cannot be physically implemented exactly. For such problems, a solution based on nominal values of the parameters may deviate severely from feasibility or optimality.
In robust optimization one is looking for a solution which satisfies the actual constraints for all possible realizations of the data within a fixed uncertainty set. This new problem is called the robust counterpart of the original problem. The robust counterpart is usually a semi-infinite optimization problem and in many cases can not be converted to explicitly convex optimization programs, solvable by high performance optimization techniques. For these cases the concept of an approximate robust counterpart was proposed by Ben-Tal and Nemirovski . A solution of the approximate robust counterpart is always a robust solution of the original uncertain problem, and as shown in [1, 4], it can often be computed efficiently.
An increasingly active and challenging research area in growth and productivity studies is the nonsmooth frontier problems under a certain production technology as characterized with a production function, , , where is a scalar output and is a given set of input possibilities. The non-smooth frontier problems are triggered by facts, as first noted by Solow , the production growth in many industries and countries follows a lagging and non-smooth (e.g., stepwise) trajectory as a function of production inputs, such as investments in IT capital and human capital. The interests in the study of nonlinear and non-smooth frontiers were further intensified by the reports at an FOMC meeting in late 1996 that the negative trends in measured productivity observed in many services industries seemed inconsistent with the fact that they ranked among the top computer-using industries (Carrado and Slifman ). Similarly, the issue of the nonsmoothness has been encountered in logistics and supply-chain systems. For example, differing container/trailer standards and heterogeneous cargo-handling characteristics of ports have been identified as a major issue of efficiency in ocean transport and short sea shipping as well (Perakis and Denisis ). In a recent econometric study on port logistics by Yan et al. , it is empirically identified that some heterogeneous and time-variant production factors, such as geographic location and business structure, affect significantly the efficiency of global container ports. With the devoted efforts over the past two decades, productivity researchers have collectively identified two key characteristics of sources of non-smooth growth, namely, degenerative input and nonlinear production cost. A degenerative input is defined as the input that generates stepwise output, and nonlinear production cost is expressed in the form of , where the nonlinear term represents intangible cost, in addition to the standard linear cost and is the vector of random coefficient. It shall be noted that production frontier problems studied so far assume a linear cost structure, that is, . Under the aforementioned nonlinear cost, a nonlinear production frontier problem is constructed herein in the form of a nonlinear-cost minimization problem:
We define the nominal problem to be problem (1.1) when the random coefficient takes value equal to its expected value . In order to protect the solution against infeasibility of problem (1.1), we may formulate the problem using chance constraint as follows: It is well known that such chance constraint is nonconvex and generally intractable. However, we will solve a tractable problem and obtain a robust solution that is feasible to the chance constraint problem (1.2) when is very small and without having to increase the objective function excessively. In order to address problem (1.1), Ben-Tal and Nemirovski [1, 3] and independently by El-Ghaoui et al. [8, 9] propose to solve the following robust optimization problem where is a given uncertainty set. The motivation for solving problem (1.3) is to find a solution that immunizes problem (1.1) against parameter uncertainty. That is, by selecting appropriate set , we can find a solution to problem (1.3) that gives guarantee in problem (1.2). However, this is done at the expense of increasing the achievable objective. It is important to note that we describe uncertainty in problem (1.3) (using the set ) in a deterministic manner. In selecting uncertainty set , we feel that two criteria are important. (a)Preserve the computational tractability both theoretically and most importantly practically of the nominal problem. From a theoretical perspective it is desirable that if the nominal problem is solvable in polynomial time, then the robust problem is also polynomially solvable. (b)Being able to find a guarantee on the probability that the robust solution is feasible, when the uncertain coefficients obey some natural probability distributions.
Our goal in this paper is to address (a) and (b) above for robust nonlinear production frontier problem. Specially, we propose a new robust counterpart of problem (1.1) that has the following properties. (a) It inherits the character of the nominal problem; (b) under reasonable probabilistic assumptions on data variation we establish probabilistic guarantee for feasibility that lead to explicit ways for selecting parameters that control the robustness. The structure of this paper is as follows. In Section 2, we describe the proposed robust model and in Section 3, we show that the robust nonlinear production frontier problem can be presented in a tractable manner. In Section 4, we prove probabilistic guarantee for feasibility for this problem.
In this section, we outline the ingredients of the proposed framework for robust nonlinear production frontier problem.
2.1. Model for Parameter Uncertainty
The model of data uncertainty we consider is where is the nominal value of the data, , is a direction of data perturbation, and , are independent and identically distributed random variables with mean equal to zero, so that . The cardinality of may be small, modeling situations involving a small collection of primitive independent uncertainties, or large, potentially as large as the number of entries in the data. In the former case, the elements of are strongly dependent, while in the latter case the elements of are weakly dependent or even independent (when is equal to the number of entries in the data). The support of , can be unbounded or bounded. Ben-Tal and Nemirovski  and Bertsimas and Sim  have considered the case that is equal to the number of entries in the data.
2.2. Uncertainty Set and Related Norm
In the robust optimization framework (1.3), we consider the uncertainty set as follows: where is a parameter, which Bertsimas and Sim  have showed, related to the probabilistic guarantee against infeasibility. Bertsimas and Sim  also restricted the vector norm by imposing the condition: where if and they called such norm the absolute norm. Given a norm , we consider the dual norm defined as
We next show some basic properties of norms satisfying (2.3), which we will subsequently use in our development.
Proposition 2.1 (see ). The absolute norm satisfies the following.(1); (2)for all , such that , ; (3)for all , such that , .
2.3. The Class of Functions
We impose the following restrictions on the function in problem (1.1).
Assumption 2.2. The function satisfies (a)the function is convex in for all ; (b), for all , . Let the functions satisfy Assumption 2.2. Then it is easy to see that
3. The Proposed Robust Framework and Its Tractability
Specifically, under the model of data uncertainty in (2.1), we propose the constraint for controlling the feasibility of stochastic data uncertainty in constraint (1.3) as follows: where and the norm satisfies (2.3). We next show that under Assumption 2.2, (3.1) implies the classical definition of robustness: where is defined in (2.2). Moreover, if the function is linear in , then (3.1) is equivalent to (3.3).
Proposition 3.1. Suppose the given norm satisfies (2.3). (a)If , then satisfies (3.1) if and only if satisfies (3.3). (b)Under Assumption 2.2, if is feasible in problem (3.1), then is feasible in problem (3.3).
Proof. (a) Under the linearity assumption, (3.1) is equivalent to
while (3.3) can be written as
Suppose is infeasible in (3.5). Then there exists , such that
For all , let and . Clearly, and since , we have from (2.3) that
Hence, is infeasible in (3.4) as well.
Conversely, suppose is infeasible in (3.4), then there exists and such that For all , we let and we observe that . Therefore, for norms satisfying (2.3), we have and hence, is infeasible in (3.5).
(b) Suppose is feasible in problem (3.1). Then From (2.5) and Assumption 2.2 for all and . In the proof of part (a), we showed that is equivalent to Thus satisfies (3.3). This completes the proof.
3.1. Tractability of the Proposed Framework
Proof. (a) First, we introduce the following problems:
In , Bertsimas and Sim showed that . Therefore, we observe that
and using the definition of dual norm, , we obtain that
and so (3.14) follows. Note that
since otherwise there exists an such that , that is,
From Assumption 2.2(b) , contradicting the convexity of (Assumption 2.2(a)).
Suppose that is feasible in Problem (3.14). Defining and , we can easily check that are feasible in problem (3.16). Conversely, suppose that is infeasible in (3.14), that is, Since we apply Proposition 2.1(b) to obtain and so Thus, is infeasible in (3.16).
(b) It is immediate that (3.14) can be written in the form of (3.16).
This completes the proof.
Remark 3.3. In (1.1), let and , . Then the optimization problem (1.1) will be in the conic form: where, for every is (i)either a nonnegative orthant (linear constraints), (ii)or the Lorentz cone (conic quadratic constraints), (iii)or a semidefinite cone —the cone of positive semidefinite matrices in the space of symmetric matrices (Linear Matrix Inequality constraints). The class of problems which can be modeled in the form of (3.16) is extremely wide (see, e.g., [1–5]). It is also clear what is the structure and what are the data in (3.25)—the former is the design dimension , the number of conic constraints , and the list of the cones , while the latter is the collection of matrices and vectors of appropriate sizes. Thus, an uncertain problem of (3.16) is a collection, of instances (3.16) of a common structure and data varying in a given set . By specifying reasonable uncertainty sets in specific applications of , Ben-Tal and Nemirovski [1–5] reformulate (3.26) as a computational tractable optimization problem, or at least approximate (3.26) by a tractable problem. Therefore, Theorem 3.2 generalizes model (3.16) from optimization problem with linear object function to a more widely nonlinear object function optimization problem.
3.2. Representation of the Function
The function naturally arises in Theorem 3.2. Recall that a norm satisfies , , , and implies that . We show next that the function satisfies all these properties except the last one, that is, it behaves almost like a norm.
Proposition 3.4. Under Assumption 2.2, the function has the following properties: (a); (b); (c).
Proof. (a) Suppose there exists such that , that is, and . From Assumption 2.2(b) , contradicting the convexity of (Assumption 2.2(a)).
(b) For , we apply Assumption 2.2(b) and obtain Similarly, if we have
(c) Using (2.5) we obtain This completes the proof.
Note that the function does not necessarily define a norm for , since does not necessarily imply . However, for specific direction of data perturbation, , we can map to a function of norm such that where is linear in .
We consider the robust quadratically frontier problem, that is, . Then it is easy to see that , , , and the uncertainty set It follows that and so By Theorem 3.2, we know that the robust quadratic frontier problem is equivalent to the following tractable optimization problem where .
4. Probabilistic Guarantee
In this section, we derive a guarantee on the probability that the robust solution is feasible, when the uncertain coefficients obey some natural probability distributions. An important component of our analysis is the relation among different norms. We denote by the inner product on a vector space, , or the space of by symmetric matrices, . The inner product induces a norm . For a vector space, the natural inner product is the Euclidian inner product, , and the induced norm is the Euclidian norm .
We analyze the relation of the inner product norm with the norm defined in (3.32). Since and are valid norms in a finite dimensional space, there exist finite such that for all in the relevant space.
Proof. From the assumptions, it is easy to see that This completes the proof.
We are naturally led to bound the probability . When we use -norm in (3.2), that is, , we have
By the similar method used in Bertsimas and Sim , we can get the following result which provides a bound that is independent of the solution .
The authors are grateful to the editor and the referees for their valuable comments and suggestions. This work was supported by the Key Program of NSFC (Grant no. 70831005) and the National Natural Science Foundation of China (11171237, 11126346).
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