Computing Shadow Prices with Multiple Lagrange Multipliers

JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2020070 MANAGEMENT OPTIMIZATION Volume 17, Number 5, September 2021 pp. 2307{2329 COMPUTING SHADOW PRICES WITH MULTIPLE LAGRANGE MULTIPLIERS Tao Jie and Gao Yan∗ 516 Jungong Road Business School, University of Shanghai for Science and Technology Shanghai 200093, China (Communicated by Liqun Qi) Abstract. There is a wide consensus that the shadow prices of certain resources in an economic system are equal to Lagrange multipliers. However, this is misleading with respect to multiple Lagrange multipliers. In this paper, we propose a new type of Lagrange multiplier, the weighted minimum norm Lagrange multiplier, which is a type of shadow price. An attractive aspect of this type of Lagrange multiplier is that it conveys the sensitivity information when resources are required to be proportionally input. To compute the weighted minimum norm Lagrange multiplier, we propose two algorithms. One is the penalty function method with numeric stability, and the other is the accelerated gradient method with fewer arithmetic operations and a convergence rate of O( 1 ). Furthermore, we propose a two-phase procedure to compute k2 a particular subset of shadow prices that belongs to the set of bounded La- grange multipliers. This subset is particularly attractive since all its elements are computable shadow prices. We report the numerical results for randomly generated problems. 1. Introduction. In recent years, given that the importance of environmental problems and resource allocation issues appear to be growing by the day, shadow prices (hereafter referred to as SP) have received significant attention from researchers, particularly in the areas of the energy economy [17, 19, 15], resource pricing [9] and decision-making analysis [14,7,8]. A shadow price, as stated by Tinbergen and Kantorovich respectively in the 1930s, is denoted as the infinitesimal change in the utility arising from an infinitesimal change in the resources constraint (see [21]). In general, a Lagrange multiplier vector (hereafter referred to as an LM) expresses the sensitivity information as the SP only when the LM is unique. In multiple LM cases, researchers have long recognized that multiple LMs may lead to incorrect computations of SPs. This observation raises the natural question of what kind of sensitivity information do Lagrange multipliers convey when they are not unique [5]. As to linear programming models, Jansen et al. [13] pointed out that: managers 2020 Mathematics Subject Classification. Primary: 90C25, 90C30; Secondary: 90C31. Key words and phrases. Shadow Price, Multiple Lagrange Multipliers, Weighted Norm La- grange Multiplier, Accelerated Gradient Method, Convex Programming. Tao Jie is supported by National Natural Science Foundation of China grant No. 71601117 and Soft Science Foundation of Shanghai grant No. 19692104600. ∗ Corresponding author: Gao Yan. 2307 2308 TAO JIE AND GAO YAN who build their own microcomputer linear programming models are apt to misuse the resulting SPs and shadow costs. Fallacious interpretations of these values can lead to expensive mistakes, especially unwarranted capital investments. Aucamp and Steinberg [2] also pointed out that the inequivalence of LMs and SPs occurs when the optimal solution is degenerated, which is also shown in [1, 12, 23]. In [2], Aucamp and Steinberg also proposed the notion of selling shadow prices and buying shadow prices, which correlate to two particular types of LMs. Akgul [1] also proposed a tableau - based method to compute these two types of SPs using linear programming models. Gauvin [11] extended the notion of selling=buying shadow prices to nonconvex programming models by resorting to parametric programming methods. However, a major drawback of these studies is that they fail to offer a path to compute the SPs in nonlinear programming models. To address this problem, Bertsekas et al. [6] proposed that the minimum norm LM can be used as a kind of SP in general programming models. The proposition of the minimum norm LM is of great significance since it provides a way to compute a certain SP. However, the lack of managerial significance of the minimum norm LM limits its application value. Furthermore, selling = buying shadow prices may also be impractical in real- world applications. To see this, in the production process, resources are usually proportionally input, and thus it is unrealistic to consider the increment of each resource separately. In this research, we first extend the work of [6], and introduce the notion of the weighted minimum norm Lagrange multiplier, which is a new type of shadow price. In practice, managers have the flexibility to adjust the weights of the resources being input. More importantly, the weighted minimum norm Lagrange multiplier is computable in the sense that two algorithms are proposed to compute this type of LM, and the convergence results of these two algorithms are also proved. Moreover, we propose a two-phase procedure to compute the set of shadow prices that belongs to a bounded subset of Lagrange multipliers. This set is of particular interest since the elements in this set are computable shadow prices. We also note that, to the best of our knowledge, the two-phase procedure is the first algorithm to identify whether a certain LM is a SP. 1.1. Structure of the paper. In Section2, we provide several relevant definitions, such as Lagrange multipliers and shadow prices. In Section3, we introduce the notion of the weighted minimum norm Lagrange multiplier, and discuss the managerial significance of this type of Lagrange multiplier. In Section4, we propose two algorithms to compute the weighted minimum norm Lagrange multiplier: one is the penalty function method with numeric stability, and the other is the accelerated gradient method with fewer arithmetic operations and a convergence rate of 1 O( k2 ). In Section5, enlightened by Wei and Yan's vertex enumeration method [22], we propose a two-phase procedure to compute the set of shadow prices that belongs to a bounded subset of Lagrange multipliers. This subset is particularly attractive since all elements in this set are computable shadow prices. In Section6, numerical examples are shown to testify the computational efficiency of the algorithm herein. Finally, some concluding remarks are given in Section7. 2. Preliminaries. 2.1. Notations. Unless particularly specified, the norm used in this paper is the n n n n Euclidean norm; R denotes the n - dimensional Euclidean space; S , S+ and S++ COMPUTING SHADOW PRICES 2309 denote the space of symmetric matrix, positive semi-definite matrix and positive definite matrix respectively; ·!· denotes the limit of a certain sequence; g+ = maxf0; gg; e denotes the vector whose elements are all equal to 1; ej represents the vector whose elements are all equal to zero except the jth element being one; r n denotes the gradient operator; let f : R ! R, rif denotes the ith element of rf; For the convex function f and a certain point x, @f(x) denotes the subgradient of f at x; For the sets A and B, AnB represents A T Bc; For a given matrix D, D(m; n) represents the element on the mth row and nth column of D. D(m; :) represents the mth row of D, and D(1 : m; :) represents the matrix constituted by the first m rows of D. 2.2. Basic definitions. Consider the following economic system: max f(x) (1) s.t. gi(x) ≤ 0; i = 1; : : : ; m; n where f; gi; i = 1; : : : ; m, are continuously differentiable functions defined on R , and x 2 Rn is the decision variable. The objective function f represents the ben- efit function, and gi; i = 1; : : : ; m represents the constraint on the supplement of resources. Suppose x∗ is a locally optimal solution of Problem (1), we state that an ∗ ∗ ∗ inequality gi(x) ≤ 0 is active at x if gi(x ) = 0. We use I(x ) to denote the set of ∗ ∗ ∗ indices of active inequalities at the optimal solution x , i.e., I(x ) = fi j gi(x ) = 0g. ∗ c ∗ ∗ c ∗ Correspondingly, I(x ) is the complement of I(x ), i.e., I(x ) = fi j gi(x ) < 0g. The economic significance of the model is to make the best decision in order to maximize the value function under the constraint of limited resources. Let X = fx j gi(x) ≤ 0; i = 1; : : : ; mg be the feasible set of Problem (1), a vector y 2 Rn is said to be a tangent direction of X at x∗ 2 X if either y = 0 or there ∗ exists a sequence fxkg ⊆ X such that xk =6 x for all k and ∗ ∗ xk − x y xk ! x ; ∗ ! : kxk − x k kyk The set of all tangent directions of Problem (1) at x∗ is called the tangent cone ∗ ∗ ∗ of Problem (1) at x , denoted by TX(x ). A cone V (x ) is called the subspace of ∗ ∗ T first order feasible variations if V (x ) = fy j rgi(x ) y ≤ 0g. The Lagrangian of m P Problem (1) is defined as L(x; µ) = f(x) − µjgj(x). j=1 An LM of the nonlinear optimization problem is defined as follows [5]: Definition 1. Suppose x∗ is a locally optimal solution of Problem (1). If there exists µ∗ 2 Rm satisfying: m ∗ P ∗ ∗ rf(x ) − µj rgj(x ) = 0; j=1 µ∗ ≥ 0; ∗ ∗ c µi = 0; i 2 I(x ) ; then µ∗ is called the Lagrange multiplier of Problem (1) at x∗. Definition 2. Let L > 0. A function f : Rn ! R is said to be L - smooth over a set D ⊆ Rn if it is differentiable over D and satisfies krf(x) − rf(y)k ≤ Lkx − yk; for all x; y 2 D. The constant L is called the smoothness parameter. 2310 TAO JIE AND GAO YAN 2.3.

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