55 Dual problem minimization with problem constraints of the form55双的问题最小化的约束问题 25页PPT
MKL_Feb2014
Cortes et al
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Decision function
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Structured text 3.2M constant and variable features
Real Time processing Using small number of features for prediction speed
Outline
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Kernel Combination
1. 2.
Heterogeneous Information Fusion Feature Selection
非内点同伦方法求解双层规划问题
i=1
Ω
w ∈ Rn+m × Rm ++2 : g˜(w) ≤ 0 , Ω 0
w ∈ Rn+m × Rm ++2 : g˜(w) < 0 ,
m1
Ω1 Ω × Rm ++1 × Rm+m2 , Ω10 Ω 0 × Rm ++1 × Rm+m2 , ∂Ω
w ∈ Ω : g˜i(w) = 0 ,
i=1
I(w) {i ∈ {1, · · · , m1} : g˜i(w) = 0} , IG(x, y) {i ∈ {1, · · · , m2} : Gi(x, y) = 0} .
§1 引 言
双层规划问题(BLPP)是一种优化问题, 它受到另一层优化问题的约束. 其通用形式如下: minx,y f (x, y) s.t. g(x, y) ≤ 0, miny F (x, y) s.t. G(x, y) ≤ 0,
其中(x, y) ∈ Rn+m和f, F : Rn+m → R, g = (g1, · · · , gm1 )T : Rn+m → Rm1 , G = (G1, · · · , Gm2 )T : Rn+m → Rm2 .
当t = 0时, 方程(2)对应于BLPP∇的f˜(KwK) T+系∇统g˜(:w)v + ∇wh(w, t)λ = 0
h(w, t) = 0 V g˜(w) = 0, v ≥ 0, g˜(w) ≤ 0.
为方便起见, 将H(θ, θ(0), t), 表示为Hθ(0) , 同伦方程的解集用下式表示
H −1
双 层 规 划 也 称 为 双 层 优 化, 是 指 模 型 约 束 中 包 含 子 目 标 函 数 和 子 约 束 条 件 的 数 学 规 划. Bracken和McGill[1]在研究不平衡经济市场竞争中首次提出的. Cander和Norton[2]在科学报 告中正式提出了双层规划. 自1980年以来, 双层规划在理论和算法上得到了广泛的关注和迅速 的发展, 并逐渐形成了一个新的运筹学分支. 目前, 双层规划问题在工程设计, 生态, 经济等领域 有着广泛的应用, 例如农业信贷分配[3], 经济规划[4], 交通运输[5], 优化交通信号[6-7]等.
LectureF 17_MATH 367
Subject to x j ∈ {0,1} , for all j . This problem can be solved easily, and we obtain
g ( λ ) = v ( Pλ ) = λW + ∑ j min ( 0, c j − λ w j ) . To solve the Lagrangean dual problem requires to find λ ≥ 0 , which satisfy the following expression g ( λ ) = v ( Pλ ) , and maximize the objective function.
P : minimize f ,
subject to
∑
j
aij x j ≥ bi , for i = 1, 2,..., m , x = ( x1 , x2 ,..., xn ) ∈ Ω .
Байду номын сангаас
(17.1)
Here the restriction “ x ∈ Ω ” covers all the constraints. Of course several other types of constraints can be appeared (for instance several kinds of equalities, inequalities, integrality etc).
Solution: The knapsack constraint (17.3) may be included into the objective function,
with (the single) multiplier λ , yielding. Pλ : minimize
高级微观经济学 (厦门大学)chp.3
1
input
x = (x1; x2; :::; xn); x 0
y = f (x); y 0
Assumption: Production f : R+n ! R+ is continuous, strictly increasing and strictly quasiconcave on R+n ; f (0) = 0:
Remark: if the production function is quasiconcave, ij 0
2
the closer it is to zero, the more substitution between the inputs. (…g. 3-2, 121)
Example: Constant elasticity substitution (CES) production function y = (x1 + x2)1=
Marginal product: Isoquant:
@f (x) @xi
Q(y) = fx 0jf (x) = yg
Remark: if the production function is quasiconcave, the isoquant is convex.
Marginal rate of technical substitution(MRTS)
if the …rm’s objective is to maximize pro…t, it should be cost minimizing.
price vector.
w = (w1; w2; :::; wn)
Lingo solution report中各项的含义
即以此功德，庄严佛净土。
上报四重恩，下救三道苦。
惟愿见闻者，悉发菩提心。
在世富贵全，往生极乐国。
（一）优化模型的组成优化模型包括以下3部分：l Objective Function：目标函数是一个能准确表达所要优化问题的公式。
l Variables：Decision variables（决策变量），在模型中所使用的变量。
l Constraints：约束条件。
（二）Lingo软件使用的注意事项（1）LINGO中不区分大小写字母，变量（和行名）可以使用不超过32个字符表示，且必须以字母开头。
（2）在命令方式下（Command Window中），必须先输入MODEL：表示开始输入模型。
LINGO中模型以“MODEL:”开始，以“END”结束。
对简单的模型，这两个语句也可以省略。
（3）LINGO中的语句的顺序是不重要的，因为LINGO总是根据“MAX=”或“MIN=”语句寻找目标函数，而其它语句都是约束条件（当然注释语句和TITLE除外）。
（4）LINGO模型是由一系列语句组成，每个语句以分号“;”结束。
（5）LINGO中以感叹号“!”开始的是说明语句（说明语句也需要以分号“;”结束）。
（6）LINGO中解优化模型时假定所有变量非负（除非用限定变量取值范围的函数@free或@sub或slb另行说明）。
（7）当您要判断表达式输入是否有错误时，也可以使用菜单“Lingo“的”Picture“选项。
(8) 用命令"@BND(下界, 变量名, 上界)"设置变量的上界和下界(9) 用命令"@free(x1)"取消变量x1的非负限制，x1可以取正实数和负实数(10)一般整数变量可以用"@GIN(变量名)"来标识，0-1型变量可以用"@BIN(变量名)"来标识（三）Solution Report各项的含义例1 将以下模型粘贴到Lingo中求解，其中第一行MODEL和最后一行END在Lingo Model 窗口下可以不要。
后格莱斯理论之列文森
Types of Q-implicature
Q-scalar implicature Q-clausal implicature (p or q,p and q,since p then q,if p then q,know that,believe that) Q-alternate implicature,which come from a nonentailment semantic (contrast) set.
<succeed,try˃ In 1888,Van Gogh tried to set up an artist's studio at Arles. +˃In 1888,Van Gogh did not succeed in setting up an artist's studio at Arles. <boil,grill,stir-fry˃ Stir-fry the bean sprouts,please. +˃please,don't,for example,boil the bean spouts as well. (Yan Huang2009:44)
In Sperber and Wilson's approach and that of Horn's,it is exactly adopted what grice had anticipated--to use the principle of revelance to subsume the second maxim of Quantity.Now Levinson believes that is mistaken.In his view, maxims of Quantity have to do with the quantity of information,while the maxim of Relation is “a measure of timely helpfulness with respect to interactional goals”(p.401).and “is largely about the satisfaction of others' goals in interaction,and the satisfaction of topical and sequencing constraints in discourse,as in the expection that answer will follow a question”(Levinson 1989:467).It is not,at least not primarily,about information.So he renames the second maxim of Quantity the Principle of Informativeness,I-principle for short;and the first maxim of Quantity the Principle of Quantity,or Q-principle.
PROBLEM A HIERARCHICAL SOLUTION PROCEDURE
ASAC 2005 Satyendra KumarToronto, Ontario Faculty of Administration,University of New Brunswick, Fredericton, CanadaV. Venkata RaoDevanathTirupatiIndian Institute of Management, Ahmadabad, IndiaMULTI-PRODUCT, MULTI-CONSTRAINT SINGLE-PERIOD INVENTORYPROBLEM: A HIERARCHICAL SOLUTION PROCEDUREThis paper presents a hierarchical solution procedure of a multiple product, multipleconstraint, single period inventory problem. The hierarchical procedure decomposes theproblem into a number of sub-problems, each sub-problem corresponding to one of theconstraint sets. Each sub-problem is solved optimally by applying Lagrange multipliersand satisfying Kuhn-Tucker conditions.IntroductionA single period inventory model, also popularly known as the Newsboy or Christmas treeproblem, is to decide the quantity of a single product to be manufactured or purchased for a singleperiod such that the quantity maximizes (minimizes) the expected profit (cost) under stochasticdemand. The earliest works in this area were those by Hadley and Whittin (1963) and Hodges andMoore (1970). The later authors solved the product-mix problem with stochastic demandcompeting for a number of limited resources. Among the many extensions to the classical singleperiod inventory problems developed after the 80’s, the optimization techniques to solve multi-product, multi-constraint single period inventory problem are of relevance to the present work.Khouja (1999) in his review paper has classified the above extensions into eleven categories. Tothese, one can add another category, namely the application of techniques like marginal analysis[Hodges and Moore (1970)], Lagrange multipliers [Karmarkar (1981), and [Lau and Lau (1995),etc.], heuristics [Nahmias and Schmidt (1984), etc.], and analytical solution procedures [Ben-Daya and Raouf (1993), etc.] to solve the constrained newsboy problem.Several authors including [Hadley and Whittin (1963), Nahmias and Schmidt (1984), Lauand Lau (1995), Lau and Lau (1996), and Vairaktarakis (2000), etc.] solved the single-periodmulti-product constrained inventory models. The constraint sets often considered by these authorswere storage space, production capacity, and budget. The methodologies used to solve theseproblems were quite different. Hadley and Whittin (1963) have solved a single constraint setproblem by Lagrangian multipliers and the solution procedure was suitable for large orderquantities. They have adopted marginal analysis approach to solve for problems with small orderquantities. Nahmias and Schmidt (1984) pointed out that Lagrangian method might require morecomputation than marginal analysis. They have developed four different heuristics to solve a single constraint problem. Lau and Lau (1996) have also solved multiple constraint problems. They converted an N-variable ‘primal problem’ to an ‘M’ variable ‘dual problem’ and developed an ‘active set methods’ to solve the problem. The solution procedure developed by them would perform well only with a small number of dual variables i.e. small numbers of primal constraints. But, there are many situations where the number of constraints is large and the active set method may fail to provide an efficient solution. Recently, Layek and Montanari (2004) have considered multi-product newsboy problem with two constraints and solved the problem to minimize the sum of expected over-stocking and under-stocking cost and the resource cost dependent on the quantity.In this paper, we formulate a single-period multiple product inventory problem with a large number of constraints, and develop a hierarchical method for its solution. Section 2 describes the problem formulation, Section 3 deals with the hierarchical solution procedure with a detailed example of convergence, Section 4 provides computational experiments for convergence, and finally Section 5 summarizes the paper.Problem formulationWe came across a practical multi-product, multi-constraint problem while dealing with a dairy firm in a large city. The dairy produces four different kinds of milk and sells them through more than 100 retail outlets in the city. All these retail outlets face stochastic demand for the different types of milk. These outlets meet the demand from their stock, which is replenished once in a day. In addition, the outlets have a storage space constraint. The retail outlets sometimes face stock-outs and some times excess inventories, causing the system to incur both over-stocking and under-stocking costs. Furthermore, the quantity of each type of milk that the dairy can supply during a day is limited.NotationThe notation used in the model formulation is as follows:r: Retailer index (1 to R)p: Product index (1 to P)x: A random variable representing the demandf rp(x): Probability density-function of demand of product ‘p’ at retail outlet ‘r’.US p: Understocking cost of product 'p' (INR per unit)OS p: Overstocking cost of product 'p' (INR per unit)Cap r: Storage capacity at retail outlet 'r' (units)Sup p: Available supply of product 'p' (units)Q rp: Quantity of product ‘p’ to be supplied at retail outlet ‘r’ (decision variable)E rp: Expected cost of product 'p' at retail outlet 'r' when supply quantity is Q rp.Objective functionThe objective is to minimize the total expected under-stocking and over-stocking cost for all products at all retail outlets as defined below in the primal problem.Z 1: Minimize (Qrp)R P ∑∑E rp (Q rp )r =1 p =1R s.t. ∑Q rp −Sup p ≤0 p =1,..., P r =1P ∑Q rp −Cap r ≤0 r =1,..., R(P)p =1 Q rp ≥0 r =1,..., R & p =1,..., PQ rp∞ where , E rp (Q rp ) =OS p ∫(Q rp −x ). f rp (x ).dx +US p ∫(x −Q rp ). f rp (x ).dx0 Q rp The constraints of the problem are: a) the supply constraint of each product, b) the storage space constraint of each retail outlet, and c) the non-negativity constraints. The storage space constraint for each outlet arises because of limitations of physical and refrigerator space at the outlet.Solution methodologyThe above-mentioned problem has a convex objective function [Federgruen and Zipkin (1984)] with linear constraints. Thus, the objective function and each of the constraints are differentiable functions, and there will be no duality gap in the Lagrangian dual. We can solve this problem optimally by relaxing the constraints by Lagrange multipliers and then applying the Kuhn-Tucker conditions [Shapiro (1979)]. Suppose all constraints except the non-negativity restrictions are placed in the Lagrangian. Then the dual problem becomes:max Θ(δ, λ)suchthatδ =(δ1,...,δp )≥0 Lagrange multiplier for sup ply constra int λ =(λ1,..., λr )≥0 Lagrange multiplier for storage constra int(D) where ,⎧R P P R R ⎛ P ⎞ ⎫Θ(δ, λ)=inf ⎨∑∑E (Q )+ ∑δ ⎜⎛∑Q −Sup ⎟⎞+∑λ ∑Q −Cap rp ⎩r =1 p =1 rp rp p =1 p ⎝r =1 rp p ⎠ r =1 r ⎝⎜⎜p =1 rp r ⎠⎟⎟Q ≥0∀r , p ⎬⎭The infimum of the dual problem is attained at⎧0, for δp +λr ≥US p Q rp = ⎪⎨⎪⎪F rp −1 ⎜⎛ US p −δ p −λr ⎟⎞ , otherwise . (1)⎪⎝⎜US p + OS p ⎠⎟⎩ Thus, the values of Q rp can be easily calculated for any given values of the dual variables. Also, the dual problem is a concave problem and the partial derivatives of the objective function with respect to the dual variables are given by:∂Θ R =∑Q rp − Sup p ∂δp r =1(2)∂Θ P =∑Q rp − Cap r ∂λr p =1 where Q rp is given by equation 1. The optimality conditions for the dual are given by:R RIf δp > 0, then ∑Q rp − Sup p = 0, otherwise if δ p = 0, then ∑Q rp − Sup p ≤ 0r =1 r =1 (3)P PIf λr > 0, then ∑Q rp − Cap r = 0, otherwise if λr = 0, then ∑Q rp − Cap r ≤ 0p =1 p =1 The above multiple constrained problems can be optimally solved by the sub-gradient optimization technique [Shapiro (1979)]. However, using this technique to find the values of (R+P) vectors is computationally very difficult. Therefore, we have adopted a hierarchical method to solve the problem, where we have broken the problem into two sub-problems. Each sub-problem deals with a single set of constraints. The single constraint problem can be optimally solved by binary search method.Hierarchical solution procedureWorking directly with the dual problem, initially we assume that δ = 0 and λ = 0. The solution given by (1) will be optimal if it satisfies both the supply and capacity constraints. Suppose that at least one supply constraint is not satisfied, then there exists δ1 such thatIf ∑ R F rp −1 ⎜⎛ US p ⎟⎟⎞ − Sup p ≤0, then δ1 p =0 r =1 ⎝⎜US p+ OS p ⎠ (4)p p else , obtain 0<δ 1 p < US p by binary search method , suchthat ∑ R F rp −1 ⎜⎛ US −δ 1 ⎟⎟⎞ − Sup p =0 r =1 ⎝⎜ US p + OS p ⎠1The result of (4) improves the dual solution [i.e. Θ(δ ,0)≥Θ(0,0)], and by the Weak Duality Theorem provides an improved lower bound on the optimal primal objective. Now given the values of δ1 (the superscript shows the iteration number), we will compute λ such that:p p If ∑ P F rp−1 ⎜⎛ US −δ 1 ⎟⎟⎞ − Cap r ≤ 0, then λ1 r = 0 p =1 ⎝⎜ US p + OS p ⎠ (5) 1 1 −1 ⎜ p p r ⎟else , obtain 0<λr <(U S p −δ p ) b y binary search method , suchthat ∑ PF rp ⎛ US −δ 1 −λ1 ⎞− Cap r = 0 p =1 ⎜⎝ US p + OS p ⎟⎠ 1 1 1Again, we have improved the lower bound [i.e. Θ(δ , λ )≥Θ(δ ,0)]. Moreover, we have a primal feasible solution that improves an upper bound. The quantities are given by:⎧⎛ US −δ1−λ1 ⎞ 1 ⎪⎪F rp −1 ⎜⎜ p p r ⎟⎟ , if US p −δ 1 p −λ1 r > 0Q rp =⎨ ⎝ US p + OS p ⎠ (6)⎪⎪0, otherwise ⎩ The solution in (6) is primal feasible because the supply constraints are fulfilled in (4) and with λ1 ≥ 0 , there exists either of following two possibilities:R R ⎛ US ⎞1 1 −1 ⎜ p ⎟if δp = 0, then ∑Q rp − Sup p ≤∑ F rp ⎟− Sup p ≤ 0 r =1 r =1 ⎝⎜US p + OS p ⎠(7)R R ⎛ US −δ 1 ⎞1 1 −1 ⎜ p p ⎟else δ p > 0, then r ∑=1 Q rp − Sup p ≤ r ∑=1 F rp ⎜⎝ US p + OS p ⎟⎠ − Sup p = 0 Thus, the solution is feasible to supply constraints. Moreover, λ1 in (5) was obtained such that the solution is feasible with respect to the capacity constraint. The second iteration will adjust the dual variables and improve the dual solution. If δ 1p = 0 , then we need not require any∂Θ R 1 R 1adjustment because =∑Q rp − Sup p ≤ 0. If δ1 p > 0 and ∑Q rp − Sup p = 0 , again there is no need ∂δ p r =1 r =1R1for adjustment. However, if ∑Q rp − Sup p < 0 , the solution can be improved by decreasing ther =1value of δ 1p . Now, we will obtain δ p 2 according to the following: ⎛ US −λ1 ⎞ if λ1 r ∑<US p} F rp −1 ⎝⎜⎜ US p p + OS r p ⎠⎟⎟ − Sup p ≤ 0 then δ p 2 = 0 {r (8)else obtain 0<δ p 2 <δ 1 p by binary search method , suchthat ∑ F rp −1 ⎛⎜ US p −δ p 2 −λ1r ⎞⎟ − Sup p = 0 {r λ1 r <US p −δ p 2 } ⎜⎝ US p + OS p ⎟⎠2 1 11The result of (8) improves the dual solution [i.e. Θ(δ , λ )≥Θ(δ , λ)], and the lower21bound of the primal solution. Now given the values of δ2, we will compute λr ≥λr such that:If ∑ P F −1 ⎛⎜ US p −δ p 2 ⎞⎟ Cap ≤ 0, then λ2 = 0 p =1 rp⎜ US + OS ⎟− r r ⎝ p p ⎠ else , obtain λ1r <λ2 r < US p −δ p 2by binary search method , suchthat {p ∑ F rp−1⎜⎛ US p −δ p 2 −λ2 r ⎟⎟⎞− Cap r = 0 δ p 2 <US p −λ2 r } ⎜⎝ US p + OS p ⎠ (9)Again we have improved the dual solution [i.e. Θ(δ 2, λ2 ) ≥Θ(δ 2, λ1 )], and hence improved the upper bound of the primal solution by Weak Duality Theorem . Moreover, we2 12 1 2 2 11have δ ≤δ , λ ≥λ , Θ(δ , λ)≥ Θ(δ , λ), and a new primal feasible solution Q 2 given by 1 with (δ, λ) = (δ2, λ2). It is clear that this kind of behaviour will continue in subsequent iterations and finally the lower bound and the upper bound will converge and thus we will obtain the optimal solution. This is because the primal problem is a convex problem and has an optimal solution and the lower bound will monotonically increase and upper bound will monotonically decrease and finally both will converge to the optimal solution.The other consideration of solving the problem is to start with the constraint that provides higher value of total expected cost. Thus, the procedure starts with the tighter lower bound and the optimal solution may be reached in lower number of iterations. Moreover, to improve the performance of the solution procedure, we have provided a limit on the total number of iterations. So, once the maximum number of iteration will reach, the algorithm will return the primal feasible solution.Now, the next step may be to obtain the quantity in integer values. We obtain the highest integer value Q rp lower than the solution value. Then, we calculate the expected cost of each product at Q rp and Q rp +1 level. Then, we compute the difference in the expected cost [E(Q rp ) – E(Q rp +1)] and increase the value of Q rp to Q rp +1 with maximum positive difference in the expected cost. We repeat the above process until either the constraints are not violated or the change in the expected cost becomes negative. This procedure provided us the solution even when the optimal solution is not obtained till the maximum number of iterations.The hierarchical method of solving the two constraints set problem is represented in Figure 1. Here the total expected cost with δ is high, hence we have followed the iterative process as follows:Figure 1: Hierarchical method for solving two constraints set problem ExampleWe have taken an example of a two-retail outlet and two- product problem, where the demand of each product at each retail outlet is independent and normally distributed. The storage capacities at retailer-1 and retailer-2 are 40 and 45 units respectively. The supplies of product-1 and product-2 are 40 and 45 units respectively. The demand, over-stocking and under-stocking parameters are provided in the following table.R1-P1 R1-P2 R2-P1 R2-P2Meandemand 20 25 25 20deviation 2 4 3 5Standardcost 1 2 1 2Overstockingcost 4 5 4 5UnderstockingTable 1: Parameter values for example problemIn case of normal distribution, the expected cost associated with any product and any retail outlet can be calculated by equation 10 [refer Lau (1997) for derivation]. The expected cost function can be given by:E rp (Q rp ) = OS p . ⎛⎜⎜ Qrp ∫(Q rp − x ). f rp (x ).dx ⎞⎟⎟ + US p . ⎛⎜⎜∞∫(x − Q rp ). f rp (x ).dx ⎞⎟⎟⎝ 0 ⎠⎝Qrp ⎠Solving, we haveE rp (Q rp ) =σrp .[− US p .Z rp + (US p + OS p ){Z rp .Φrp (Z rp ) +φrp (Z rp )}]where , Z is the std .normal ,(10)φ is the probablity density function ,and Φ is the cumulative probablity density function . Iterative stepsTaking one constraint at a time, the total expected cost with capacity constraint is more than the total expected cost with supply constraint. Therefore, we will first solve the capacity constraint sub-problem and then solve the supply constraint sub-problem.For the given value of δ equal to zero, we solved for λ (superscript omitted) values andthe values of λ1 and λ2 satisfying equation equivalent to (4) were 3.3189 and 1.5000 respectively.Applying the dual values in equation 1, we obtained the quantity values. We havecalculated the expected cost by putting the quantity value in equation 10. The sum of the expected cost i.e. the value of lower bound was equal to 24.63.Applying the values of λ1 and λ2 equal to 3.3189 and 1.5000 respectively, we solved for δvalues and the values of δ1 and δ2 satisfying equation equivalent to (5) were 0.5860 and 0.0000 respectively.Substituting the dual values in equation 1, we obtained the quantity values. The values ofQ 11, Q 12, Q 21, and Q 22 were 15.89, 22.19, 24.11 and 20.00 respectively. We have calculated the expected cost and the sum of the expected cost i.e. the value of upper bound was equal to 34.33. This completes Iteration 1.For the given value of δ1 and δ2 equal to 0.5860 and 0.0000 respectively, we solved for λvalues and the values of λ1 and λ2 satisfying equation equivalent to (8) were 2.9915 and1.2321 respectively.Applying the dual values in equation 1, we obtained the quantity values. The value ofnew lower bound was equal to 24.76.Applying the values of λ1 and λ2 equal to 2.9915 and 1.2321 respectively, we solved for δand the values of δ1 and δ2 satisfying equation equivalent to (9) were 0.8993 and 0.0000 respectively.Substituting the dual values in equation 1, we got the quantity values. The values of Q11, Q12, Q21, and Q22 were 15.97, 22.75, 24.03 and 20.48 respectively. The value of new upper bound was 31.21. This completes Iteration 2.Table 2 presents the quantities, Lagrange multiplier, lower bound and upper bound values after each iteration.Iteration Q11 Q12 Q21 Q22 λ1 λ2 δ1 δ2 LB UB1 15.89 22.19 24.11 20.00 3.3139 1.5000 0.5860 0.0000 24.63 34.332 15.97 22.75 24.03 20.48 2.9915 1.2321 0.8993 0.0000 24.76 31.213 16.04 23.08 23.96 20.74 2.7915 1.0880 1.0894 0.0000 25.19 29.604 16.10 23.29 23.90 20.90 2.6601 1.0003 1.2125 0.0000 25.58 28.665 16.14 23.43 23.86 21.00 2.5708 0.9430 1.2955 0.0000 25.89 28.066 16.17 23.52 23.83 21.07 2.5089 0.9042 1.3528 0.0000 26.13 27.687 16.19 23.59 23.81 21.12 2.4653 0.8775 1.3929 0.0000 26.31 27.418 16.20 23.64 23.80 21.16 2.4344 0.8587 1.4212 0.0000 26.44 27.239 16.22 23.67 23.78 21.18 2.4122 0.8454 1.4415 0.0000 26.53 27.1010 16.22 23.69 23.78 21.20 2.3964 0.8359 1.4559 0.0000 26.60 27.0111 16.23 23.71 23.77 21.21 2.3851 0.8291 1.4662 0.0000 26.65 26.9512 16.23 23.72 23.77 21.22 2.3770 0.8243 1.4736 0.0000 26.69 26.9013 16.24 23.73 23.76 21.23 2.3712 0.8208 1.4789 0.0000 26.71 26.8714 16.24 23.74 23.76 21.23 2.3669 0.8183 1.4828 0.0000 26.73 26.8415 16.24 23.74 23.76 21.24 2.3637 0.8165 1.4857 0.0000 26.74 26.8316 16.24 23.75 23.76 21.24 2.3615 0.8151 1.4878 0.0000 26.75 26.8117 16.24 23.75 23.76 21.24 2.3598 0.8141 1.4893 0.0000 26.76 26.8118 16.24 23.75 23.76 21.24 2.3586 0.8134 1.4904 0.0000 26.77 26.8019 16.24 23.75 23.76 21.24 2.3578 0.8129 1.4912 0.0000 26.77 26.7920 16.24 23.75 23.76 21.24 2.3572 0.8126 1.4918 0.0000 26.78 26.78Table 2: Iterative values of convergence testThe table shows that the lower bound is monotonically increasing and the upper bound is monotonically decreasing. After 20 iterations the lower bound and the upper bound converge and provide the optimal solution. The graph showing the convergence of lower bound and upper bound is as follows:L B / U B V a l u e s Lower Bound - Upper Bound Convergence1 2 3 4 5 6 7 8 9 1011121314151617181920IterationFigure 2: Graph showing iterative results and convergenceExperimental resultsWe have implemented the hierarchical solution procedure in Visual Basic 6.0 with Excelinterface. We have designed many experiments to test the quality of the solution. The objective of the experiments is to computationally verify the convergence of lower bound and the upper bound of the problem. The parameters to measure the performance of the hierarchical method are as follows:Percentage of instances that converged under the given stopping rule.Average number of iterations required for convergence.Average percentage deviation between lower bound and upper bound.Average time required for convergence.We have run the experiments on Personal Computer having Pentium III 500 MHzprocessor and 128 MB RAM.Experiment-1In this experiment, we have considered50 retailers and 4 products.Over-stocking cost is considered 1 for all products. Under-stocking cost is considered as2.0,3.0,4.0 and5.0 for the four different products respectively.Mean demand is randomly generated between 10 and 40 for every (r, p) combination.2 different scenarios of coefficient of variation, which is randomly generated between 0.1and 0.25, and 0.1 and 0.4 for every (r, p) combination.2 different scenarios of storage space capacity, which is randomly generated between 100to 150 and 75 to 100 for every retailer.2 different scenarios of supply, which is randomly generated between 1000 to 1500 and500 to 800 for every product.Thus, we have generated 8 test problems and for each test problem we have run 25 instances. The solution algorithm was run for a maximum of 100 iterations for each problem instance. The results of these test problems are presented in the following table:Test Coefficient ofvariationStoragecapacitySupplyInstancesconvergedAv no. ofiterations%deviationAverage time(seconds)1 0.1 - 0.25 100 – 150 1000-1500 25 1.2 0.0% 0.202 0.1 - 0.25 100 – 150 500-800 25 5.8 0.0% 10.523 0.1 - 0.25 75 – 100 1000-1500 25 1.0 0.0% 2.444 0.1 - 0.25 75 – 100 500-800 24 13.5 0.5% 9.005 0.1 – 0.4 100 – 150 1000-1500 25 1.2 0.0% 0.126 0.1 – 0.4 100 – 150 500-800 25 18.8 0.0% 15.087 0.1 – 0.4 75 – 100 1000-1500 25 1.0 0.0% 1.288 0.1 – 0.4 75 – 100 500-800 23 46.7 0.76% 65.48Table 3: Convergence test results for Experiment-1In this experiment, 197 out of 200 instances have converged. The maximum average deviation is 0.76%. The average number of iterations to convergence is less than 20. The average time to solve a problem is less than 15 seconds. The average time in some tests is high as compared to the other tests because the constraints are very tight and thus the value of cumulative density function is even less than 0.1. For these low values of the cumulative density function, we have calculated the respective normal value by Microsoft Excel (instead of using inverse CDF conversion table), which has taken more time.Experiment-2This experiment has been designed to capture identical retailers facing random demand. In this experiment, we have considered50 retailers and 4 productsOver-stocking cost is considered 1 for all products. Under-stocking cost is randomly allocated between 2 to 10 for every instance.Mean demand and standard deviation is randomly generated between 10 to 30 and 1 to 10 respectively for every (r, p) combination.4 different scenarios of storage space capacity are considered and are equal to 100 / 80 /60 / 40 for all retailers. 4 different scenarios of supply are considered and are equal to1000 / 800 / 600 / 400 for all products.All other parameters are similar to the previous experiment. The results of these test problems are presented in the following table:Test StoragecapacitySupplyInstancesconvergedAverage numberof iteration% deviationAverage time(seconds)1 100 1000 25 1.00 0.0% 0.362 80 1000 25 1.00 0.0% 1.243 60 1000 25 1.00 0.0% 1.524 40 1000 25 1.00 0.0% 3.125 100 800 25 1.12 0.0% 0.766 80 800 25 1.92 0.0% 1.407 60 800 25 1.96 0.0% 4.688 40 800 25 1.40 0.0% 4.089 100 600 25 1.00 0.0% 3.0810 80 600 25 1.00 0.0% 3.2011 60 600 25 1.64 0.0% 4.2412 40 600 25 1.00 0.0% 7.6813 100 400 25 1.00 0.0% 6.3614 80 400 25 1.00 0.0% 6.4415 60 400 25 1.00 0.0% 6.7616 40 400 23 6.72 3.7% 8.95Table 4: Convergence test results for Experiment-2In this experiment, 398 out of 400 instances have converged. The average number of iterations to convergence is less than 2. The average time to solve a problem is less than 6 seconds. The average deviation of instances that did not converge is 3.7%.Summary of the experimental resultsThe summary of the experimental results of the problem is as follows:The percentage of convergence is more than 99%. If the algorithm does not converge, the maximum deviation from the lower bound is less than 4%.The number of iterations and time required for solving a problem increases with constraint tightness. All the non-converged instances were very tight in terms ofconstraint values.When we increased the number of iterations to 1000, all the instances have converged, but the time taken to obtain the solution was very high.Performance AnalysisThe performance of the hierarchical method deteriorates with the tightness of the constraints. This procedure will perform worst when the constraints value is less than 20% of the sum of unconstrained optimal quantity competing for the resource. The consequences of a decrease in the constraint values are:The convergence slows downThe percentage deviation of the upper bound from the lower bound increases Therefore, the tighter the constraints the greater the number of iterations required to achieve the desired convergence.ConclusionWe have discussed in the previous sections the formulation and solution procedure for a multi-product multi-constraint single period inventory problem with two constraint sets. The above procedure can also be used for solving problems with more than two constraints sets. The proposed hierarchical solution procedure has worked efficiently for problems with a large number of products and constraints.ReferencesBen-Daya, M. & A. Raouf, “On the constrained multi-item simple-period inventory problem,” International Journal of Operations and Production Management, 13(11), (1993), 104-112. Federgruen, A. & P. Zipkin, “A combined vehicle routing and inventory allocation problem,” Operations Research, 32(5), (1988), 1019-1037.Hadley, G. & T. M. Whittin, Analysis of inventory systems, Eaglewood Cliffs, N. J.: Prentice-Hall Inc., 1963.Hodges, S. D. & P. G. Moore, “The product-mix problem under stochastic seasonal demand,” Management Science, 17(2), (1970), 107-114.Karmarkar, U. S., “The multiperiod multilocation inventory problem,” Management Science, 29(2), (1981), 215-228.Khouja, M., “The single-period (news-vendor) problem: literature review and suggestions for future research,” Omega, 27(5), (1999), 537-553.Layek, L. A. & R. Montanari, “On the multi-product newsboy problem with two constraints,” Computer & Operations Research, 32, (2005), 2095-2116.Lau, H., “Simple formulas for the expected costs in the newsboy problem: An educational note, European Journal of Operational Research, 100(3), (1997), 557-561.Lau, H. & A. Lau, “The multi-product multi-constraint newsboy problem: Application, formulation and solution,” Journal of Operations Management, 13(2), (1995), 153-162.Lau, H. & A. Lau, “The newsstand problem: A capacitated multiple-product single-period inventory model,” European Journal of Operational Research, 94(1), (1996), 29-42. Nahmias, S. & C. P. Schmidt, “An efficient heuristic for the multi-item newsboy problem with a single constraint,” Naval Research Logistics Quarterly, 31, (1984), 463-474.Shapiro, J. F., Mathematical Programming: Structures and Algorithms, New York: John Wiley and Sons, 1979.Vairaktarakis, G. L., “Robust multi-item newsboy models with a budget constraint,” International Journal of Production Economics, 66(3), (2000), 213-226.。
运筹学课件8_单纯形法特殊形式
Why is the standard form the standard form?
8
4.6 ADAPTING TO OTHER MODEL FORMS
Maximize Z
The simplex method uses one criteria to choose entering basic variables
Tie for the leaving basic variable
Two or more basic variables reach zero simultaneously as the entering basic variable is increased in value
A basic variable is zero in a BFS
4.5 TIE BREAKING IN THE SIMPLEX METHOD
Different Situations Encountered with the Simplex Method
So far all examples have started in the standard form and have had a single optimal solution Other possibilities
<= constraints, Non-negativity constraints
Starting the Simplex method with an initial basic feasible solution is easy – What is this initial BFS?
9
What Constitutes Non-standard Form?
凸优化、半定规划相关Matlab工具包总结(部分为C C++)
SoftwareFor some codes a benchmark on problems from SDPLIB is available at Arizona State University.∙CSDP 4.9, by Brian Borchers (report 1998, report 2001). He also maintains a problem library, SDPLIB.∙CVX, version 1.1, by M. Grant and S. Boyd.Matlab software for disciplined convex programming.∙DSDP 5.6 , by S. J. Benson and Y. Ye, parallel dual-scaling interior point code in C (manual); source and excutables available fromBenson's homepages.∙GloptiPoly3, by D. Henrion, J.-B. Lasserre and J. Loefberg;a Matlab/SeDuMi add-on for LMI-relaxations of minimization problemsover multivariable polynomial functions subject to polynomial or integer constraints.∙LMITOOL-2.0 of the Optimization and Control Group at ENSTA.∙MAXDET, by Shao-po Wu, L. Vandenberghe, and S. Boyd. Software for determinant maximization. (see also rmd)∙NCSOStools, by K. Cafuta, I. Klep, and J. Povh. An open source Matlab toolbox for symbolic computation with polynomials in noncommutingvariables, to be used in combination with sdp solvers.∙PENNON-1.1 by M. Kocvara and M. Stingl. It implements a penalty method for (large-scale, sparse) nonlinear and semidefiniteprogramming (see their report), and is based on the PBM method ofBen-Tal and Zibulevsky.∙PENSDP v2.0 and PENBMI v2.0, by TOMLAB Optimization Inc., a MATLAB interface for PENNON.∙rmd , by the Geometry of Lattices and Algorithms group at University of Magdeburg, for making solutions of MAXDET rigorous byapproximating primal and dual solution by rationals and testing forfeasibility.∙SBmethod (Version 1.1.3), by C. Helmberg. A C++ implementation of the spectral bundle method for eigenvalue optimization.∙SDLS by D. Henrion and J. Malick.Matlab package for solving least-squares problems over convexsymmetric cones.∙SDPA (version 7.1.2), initiated by the group around Masakazu Kojima.∙SDPHA does not seem to be available any more (it was package by F.A. Potra, R. Sheng, and N. Brixius for use with MATLAB).∙SDPLR (version 1.02, May 2005) by Sam Burer, a C package for solving large-scale semidefinite programming problems.∙SDPpack is no longer supported, but still available. Version 0.9 BETA, by F. Alizadeh, J.-P. Haeberly, M. V. Nayakkankuppam, M. L. Overton, and S. Schmieta, for use with MATLAB.∙SDPSOL (version beta), by Shao-po Wu & Stephen Boyd (May 20, 1996). A parser/solver for SDP and MAXDET problems with matrixstructure.∙SDPT3 (version 4.0), high quality MATLAB package by K.C. Toh, M.J.Todd, and R.H. Tütüncü. See the optimization online reference.∙SeDuMi, a high quality package with MATLAB interface for solving optimization problems over self-dual homogeneous cones started byJos F. Sturm.Now also available: SeDuMi Interface 1.04 by Dimitri Peaucelle.∙SOSTOOLS, by S. Prajna, A. Papachristodoulou, and P. A. Parrilo. A SEDUMI based MATLAB toolbox for formulating and solving sums ofsquares (SOS) optimization programs(also available at Caltech).∙SP (version 1.1), by L. Vandenberghe, Stephen Boyd, and Brien Alkire.Software for Semidefinite Programming.∙SparseCoLO, by the group of M. Kojima, a matlab package for conversion methods for LMIs having sparse chordal graph structure,see the Research report B-453.∙SparsePOP, by H. Waki, S. Kim, M. Kojima and M. Muramatsu, is a MATLAB implementation of a sparse semidefinite programmingrelaxation method proposed for polynomial optimization problems.∙VSDP: Verified SemiDefinite Programmin, by Christian Jansson.MATLAB software package for computing verified results ofsemidefinite programming problems. See the optimization onlinereference.∙YALMIP, free MATLAB Toolbox by J. Löfberg for rapid optmization modeling with support for, e.g., conic programming, integerprogramming, bilinear optmization, moment optmization and sum ofsquares. Interfaces about 20 solvers, including most modern SDPsolvers.Reports on software:∙M. Yamashita, K. Fujisawa, M. Fukuda, K. Nakata and M. Nakata."Parallel solver for semidefinite programming problem having sparseSchur complement matrix",Research Report B-463, Dept. of Math. and Comp. Sciences, TokyoInstitute of Technology, Oh-Okayama, Meguro, Tokyo 152-8552,September 2010.opt-online∙Hans D. Mittelmann."The state-of-the-art in conic optimization software",Arizona State University, August 2010, written for the "Handbook ofSemidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications".opt-online∙K.-C. Toh, M. J. Todd, and R. H. Tütüncü."On the implementation and usage of SDPT3 -- a Matlab softwarepackage for semidefinite-quadratic-linear programming, version 4.0", Preprint, National University of Singapore, June, 2010.opt-online∙K. Cafuta, I. Klep and J. Povh."NCSOSTOOLS: A Computer Algebra System for Symbolic andNumerical Computation with Noncommutative Polynomials",University of Ljubljana, Faculty of Mathematics and Physics, Slovenia, May 2010.opt-online∙I. D. Ivanov and E. De Klerk."Parallel implementation of a semidefinite programming solver based on CSDP on a distributed memory cluster",Optimization Methods and Software, Volume 25, Issue 3 June 2010 , pages 405 - 420 .OMS∙M. Yamashita, K. Fujisawa, K. Nakata, M. Nakata, M. Fukuda, K.Kobayashi and Kazushige Goto."A high-performance software package for semidefinite programs:SDPA 7",Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, January, 2010.opt-online∙Sunyoung Kim, Masakazu Kojima, Hayato Waki and Makoto Yamashita."SFSDP: a Sparse Version of Full SemiDefinite ProgrammingRelaxation for Sensor Network Localization Problems",Report B-457, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, July 2009.opt-online∙K. Fujisawa, S. Kim, M. Kojima, Y. Okamoto and M. Yamashita."ser's Manual for SparseCoLO: Conversion Methods for SparseConic-form Linear Optimization Problems",Research report B-453, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama,Meguro-ku, Tokyo 152-8552 Japan, February 2009.opt-online∙Sunyoung Kim, Masakazu Kojima, Martin Mevissen, Makoto Yamashita."Exploiting Sparsity in Linear and Nonlinear Matrix Inequalities viaPositive Semidefinite Matrix Completion",Research Report B-452, Department of Mathematical and ComputingSciences, Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo 152-8552, Japan, November 2008.opt-online∙ D. Henrion, J. B. Lasserre, and J. Löfberg."GloptiPoly 3: moments, optimization and semidefinite programming", LAAS-CNRS, University of Toulouse, 2007.opt-online∙Didier Henrion and J茅r么me Malick."SDLS: a Matlab package for solving conic least-squares problems", LAAS-CNRS, University of Toulouse, 2007.opt-online∙M. Grant and S. Boyd."Graph Implementations for Nonsmooth Convex Programs",Stanford University, 2007.opt-online∙K. K. Sivaramakrishnan."A PARALLEL interior point decomposition algorithm for block-angular semidefinite programs",Technical Report, Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, December 2006. Revised in June 2007 and August 2007.opt-online∙Makoto Yamashita, Katsuki Fujisawa, Mituhiro Fukuda, Masakazu Kojima, Kazuhide Nakata."Parallel Primal-Dual Interior-Point Methods for SemiDefinite Programs", Research Report B-415, Tokyo Institute of Technology, 2-12-1,Oh-okayama, Meguro-ku, Tokyo, Japan, March 2005.opt-online∙ B. Borchers and J. Young."How Far Can We Go With Primal-Dual Interior Point Methods forSDP?",New Mexico Tech, February 2005.opt-online∙H. Waki, S. Kim, M. Kojima and M. Muramatsu."SparsePOP : a Sparse Semidefinite Programming Relaxation ofPolynomial Optimization Problems",Research Report B-414, Dept. of Mathematical and ComputingSciences, Tokyo Institute of Technology, Oh-Okayama, Meguro152-8552, Tokyo, Japan, March 2005.opt-online∙M. Kocvara and M. Stingl."PENNON: A code for convex nonlinear and semidefinite programming", Optimization Methods and Software (OMS), Volume 18, Number 3,317-333, June 2003.∙Brian Borchers."CSDP 4.0 User's Guide",user's guide, New Mexico Tech, Socorro, NM 87801, 2002.opt-online∙M. Yamashita, K. Fujisawa, and M. Kojima."SDPARA : SemiDefinite Programming Algorithm PARAllel Version", Parallel Computing Vol.29 (8) 1053-1067 (2003).opt-online∙J. Sturm."Implementation of Interior Point Methods for Mixed Semidefinite and Second Order Cone Optimization Problems",Optimization Methods and Software, Volume 17, Number 6, 1105-1154, December 2002.optimization-online∙S. Benson and Y. Ye."DSDP4 Software User Guide",ANL/MCS-TM-248; Mathematics and Computer Science Division;Argonne National Laboratory; Argonne, IL; March 2002.opt-online∙S. Benson."Parallel Computing on Semidefinite Programs",Preprint ANL/MCS-P939-0302; Mathematics and Computer Science Division Argonne National Laboratory 9700 S. Cass Avenue Argonne, IL, 60439; March 2002.opt-online∙ D. Henrion and J. B. Lasserre."GloptiPoly - Global Optimization over Polynomials with Matlab andSeDuMi",LAAS-CNRS Research Report, February 2002.opt-online∙M. Kocvara and M. Stingl."PENNON - A Generalized Augmented Lagrangian Method forSemidefinite Programming",Research Report 286, Institute of Applied Mathematics, University of Erlangen, 2001.opt-online∙ D. Peaucelle, D. Henrion, and Y. Labit."User's Guide for SeDuMi Interface 1.01", Technical report number01445 LAAS-CNRS : 7 av. du Colonel Roche, 31077 Toulouse Cedex 4, FRANCE November 2001.opt-online∙Jos F. Sturm."Using SEDUMI 1.02, a MATLAB Toolbox for Optimization OverSymmetric Cones (Updated for Version 1.05)",October 2001.opt-online∙Hans D. Mittelmann."An Independent Benchmarking of SDP and SOCP Solvers",Technical Report, Dept. of Mathematics, Arizona State University, July2001.opt-online∙K. Fujisawa, M. Fukuda, M. Kojima and K. Nakata."Numerical Evaluation of SDPA",Research Report B-330, Department of Mathematical and ComputingSciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku,Tokyo 152, September 1997.ps.Z-file (ftp) or dvi.Z-file (ftp)∙L. Mosheyev and M. Zibulevsky."Penalty/Barrier Multiplier Algorithm for Semidefinite Programming:Dual Bounds and Implementation",Research Report #1/96, Optimization Laboratory, Technion, November 1996.ps-file (http)Due to several requests I have asked G. Rinaldi for permission to put his graph generator on this page. Here it is: rudy (tar.gz-file)Last modified: Tue Oct 26 15:10:14 CEST 2010。
Matlab中LMI（线性矩阵不等式）工具箱使用例子
Matlab中LMI（线性矩阵不等式）⼯具箱使⽤例⼦这⼀段被⽼板逼着论⽂开题，⾃⼰找⽅向⽐较着急，最后选择了供应链控制理论的⼀个⽅向。
我要写的论⽂，⽤到了Matlab 的LMI⼯具，以及某篇论⽂中的H-inf稳定定理。
⾃⼰好好研究了好长时间，怎么也⽆法实现该论⽂当中的算例。
研究了⼀个多⽉，⾃⼰简直就快崩溃了，也搞不定问题。
我很是怀疑⾃⼰的选题是不是正确，并且怀疑⾃⼰是不是选的太难了。
如果连论⽂中的算例都⽆法实现，如何把该模型应⽤到⾃⼰论⽂当中呢？功夫不负有⼼⼈，昨⽇我加⼊了Mathworks的Matlab的Newsgroup，结果碰见⼀⽜⼈Johan，⽴即就把论⽂中的算例给写成程序。
但是他做出的结果和论⽂仍然有差别，我仍有点不⽢⼼，⼈家的论⽂发表在Automatica上，如果连这种期刊都⽔的要命，那么就没有什么学术⽔平了。
今天中午，仍然不⽢⼼，⽼爸给我打了电话让我看红场阅兵，于是我边看PPMate边漫⽆边际的核对着⾃⼰的程序。
终于做出了和算例⼀致的结果。
我搜出来的都是⼀些简单的算例，并且机会没有中⽂教程，我在这⾥就⽃胆把⾃⼰的体会写出来，试着给⼤家提供⼀点参考。
LMI：Linear Matrix Inequality，就是线性矩阵不等式。
在Matlab当中，我们可以采⽤图形界⾯的lmiedit命令，来调⽤GUI接⼝，但是我认为采⽤程序的⽅式更⽅便（也因为我不懂这个lmiedit的GUI）。
对于LMI Lab，其中有三种求解器（solver）： feasp，mincx和gevp。
每个求解器针对不同的问题：feasp：解决可⾏性问题（feasibility problem），例如：A(x)<b(x)。
< font="">mincx：在线性矩阵不等式的限制下解决最⼩化问题（Minimization of a linear objective under LMI constraints），例如最⼩化c'x，在限制条件A(x) < B(x)下。