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3 Duality: Cost and Profit as Values of Programmes with Shadow-Price Decisions

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3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price. . .

27

programmes (CPs), expounded in, e.g., [44] and [36, Chapter 7]. The present scheme

is, however, a little different in that it starts not from a single programme, yet to be

perturbed, but from a family of programmes that depend on a set of data, whose

particular values complete a programme’s specification. So, one way to perturb

a programme is simply to add an increment to its data point, thus “shifting” it

within the given family. Some, possibly all, of the scheme’s primal perturbations

are therefore increments to some—though typically not all—of the data. The same

goes for dual perturbations.

Before the duality scheme is applied to the profit and cost programmes, it is

briefly discussed and illustrated in the framework of linear programming. A central

idea is that the dual programme depends on the choice of perturbations of the primal

programme: different perturbation schemes produce different duals. Theoretical

expositions of duality usually start from a programme without any data variables

whose increments might serve as primal perturbations: say, f .y/ is to be maximized

over y subject to a number of inequalities G1 .y/ Ä 0, G2 .y/ Ä 0; : : :, abbreviated

to G .y/ Ä 0. In such a case, any perturbations must first be introduced, and the

standard choice is to add D . 1 ; 2 ; : : :/ to the zeros on the right-hand sides

(r.h.s.’s)—thus perturbing the original constraints G .y/ Ä 0 to G .y/ Ä . The

original programme has no data other than the functions f and G themselves,

and the increments f and G (which change the programme to maximization

of . f C f / .y/ over y subject to .G C G/ .y/ Ä 0) can never serve as primal

perturbations—not even if they were taken to be linear, i.e., if f and G were a

vector and a matrix of coefficients of the primal variables y D .y1 ; y2 ; : : :/. This

is because the perturbed constrained maximand must be jointly concave in the

decision variables and the perturbations,4 but the bilinear form f y is neither concave

nor convex in f and y jointly.5

But in applications, the primal programme usually comes with a set of data that it

depends on, and increments to some of the programme’s data can commonly serve

as primal perturbations. Such data shall be called the intrinsic primal parameters ;

some or all of the other data will turn out to be dual parameters. For example, in

SRP maximization (3.1.6)–(3.1.7), the fixed-input bundle k is a primal parameter

because, since the production set Y is convex, the constrained maximand is a

concave function of .y; k; v/: it is

h p j yi

hw j vi

ı .y; k; v j Y/

where ı . ; ; j Y/ denotes the 0-1 indicator of Y (i.e., it equals 0 on Y and C1

outside of Y). By contrast, the coefficient (say, p) of a primal variable (y) is not a

4

This is equivalent to joint convexity of the constrained minimand (which is the sum of the

minimand and of the 0-1 indicator function of the constraint set). In [44] it is called “the

minimand” for brevity.

5

A linear change of variables makes it a saddle function: 4f y D . f C y/ . f C y/ . f y/ . f y/

is convex in f C y and concave in f y.

28

3 Characterizations of Long-Run Producer Optimum

primal parameter (i.e., its increment p cannot be a primal perturbation) because

the bilinear form h p j yi is not jointly concave in p and y. For these reasons, all

of the quantity data, but no price data, are primal parameters for the profit or cost

optimization programmes of Sect. 3.1. As for the production set, it cannot itself

serve as a parameter because convex sets do not form a vector space to begin with.

However, once the technological constraint .y; k; v/ 2 Y has been represented

in the form Ay Bk Cv Ä 0 (under c.r.t.s.), the matrices or, more generally, the

linear operations A, B and C are vectorial data. But none can be a primal parameter,

for lack of joint convexity of Ay in A and y, etc. Nor can A, B or C be a dual

parameter (for a similar reason). Such data variables—which are neither primal nor

dual parameters, and hence play no role in the duality scheme—shall be called tertial

parameters.

It can be analytically useful, or indeed necessary, to introduce other primal

perturbations, i.e., perturbations that are not increments to any of the data (which

are listed after the “Given” in the original programme). This amounts to introducing

additional parameters, which shall be called extrinsic ; their original, unperturbed

values can be set at zeros, as in [44]. When the constraint set is represented by

a system of inequalities and equalities, the standard “right-hand side” parameters

are always available for this purpose (unless they are all intrinsic, but this is so

only when the r.h.s. of each constraint is a separate datum of the programme and

can therefore be varied independently of the other r.h. sides). Section 3.12 shows

how to relate the marginal effects of any “nonstandard” perturbations to those of the

standard ones—i.e., how to express any “nonstandard” dual variables in terms of the

usual Lagrange multipliers of the constraints. This is useful in the problems of plant

operation and valuation, including those that arise in peak-load pricing (Sect. 5.2).6

Once a primal perturbation scheme has been fully defined, the duality framework

is completed automatically (except for the choice of the topologies and the

continuous-dual spaces in the infinite-dimensional case): dual decision variables are

introduced and paired to the specified primal perturbations (both the intrinsic and

any extrinsic ones). To re-derive the primal programme as its dual’s dual, the dual

perturbations are introduced so as to be paired with the primal variables (i.e., this

match is set up “in reverse”). The perturbed dual minimand—a function of the dual

variables, the dual perturbations and the data of the original, primal programme—is

defined in the usual way (as in [44, (4.17)] but with the primal problem reoriented

6

In this as in other contexts, it can be convenient to think of extrinsic perturbations either as

(i) complementing the intrinsic perturbations (which are increments to the fixed inputs) by varying

some aspects of the technology (such as nonnegativity constraints), or as (ii) replacing the intrinsic

perturbations with finer, more varied increments (to the fixed inputs). For example, the timeconstant capacity kÂ in (5.2.3) is an intrinsic primal parameter. The corresponding perturbation

is a constant increment kÂ , which can be refined to a time-varying increment kÂ . /. The

perturbation kÂ . / is complemented by the increment nÂ . / to the zero floor for the output rate

yÂ . / in (5.2.3). The same goes for all the occurrences of k and n in the context of pumped

storage and hydro (where

is another complementary extrinsic perturbation, of the balance

constraint (5.2.15) or (5.2.35)).

3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price. . .

29

to maximization). When all the primal perturbations are intrinsic, the resulting dual

programme is called the intrinsic dual .

Some or possibly all of the dual perturbations may turn out to perturb the dual

programme just like increments to some of the data—which are thus identified as

the intrinsic dual parameters . Any other dual perturbations are called extrinsic, and

these can be thought of as increments to extrinsic dual parameters (whose original,

unperturbed values are set at zeros). However, in the profit or cost programmes, all

the dual parameters are price data (and are therefore intrinsic).

In the reduced formulations of the profit or cost problems, some of the price

data are not dual parameters because the corresponding quantities have been solved

for in the reduction process, and have thus ceased to be decision variables: e.g.,

the variable-input price w is not a dual parameter of the reduced SRP programme

in (3.2.2) because the corresponding input bundle v has been found in SRC

minimization (and so it is no longer a decision variable). But in the full (not reduced)

formulations, all the price data are dual parameters, and thus the programme’s data

(other than the technology itself) are partitioned into the primal parameters (the

quantity data) and dual parameters (the price data).

The primal and dual optimal values can differ at some “degenerate” parameter

points (see Appendix A), but such duality gaps are exceptional, and they do not

occur when the primal or dual value is semicontinuous in, respectively, the primal or

dual parameters (Sect. 6.1). Note that both optimal values, primal and dual, depend

on the data, which are the same for both programmes. So, in this scheme, either of

the optimal values (primal or dual) is a function of both primal and dual parameters,

and so it can have two types of continuity and of derivatives (marginal values):

• continuity/derivative of Type One is that of the primal value with respect to the

primal parameters, or of the dual value w.r.t. the dual parameters;

• continuity/derivative of Type Two is that of the dual value w.r.t. the primal

parameters, or of the primal value w.r.t. the dual parameters.

This useful distinction cannot be articulated when, as in [44] and [36], the primal

and dual values are considered only as functions of either the primal or the dual

parameters, respectively.

Comments (Parameters and Their Marginal Values, Dual Programme and the

FFE Conditions, the Lagrangian and the Kuhn-Tucker Conditions for LPs)

• Let the primal linear programme be: Given any p 2 Rn and s 2 Rm , and an

m n matrix A, maximize p y over y 2 Rn subject to Ay Ä s. Here, the only

intrinsic primal parameter is the standard parameter s. There is no obviously

useful candidate for an extrinsic primal parameter, and if none is introduced,

then the dual is the standard dual LP: Given p and s (and A), minimize

s over

30

3 Characterizations of Long-Run Producer Optimum

2 Rm subject to AT D p and

0, where AT is the transpose of A.7 The

only dual parameter is p.

• If both programmes have unique solutions, yO .s; p; A/ and O .s; p; A/, with equal

values V .s; p; A/ WD p yO D O s DW V .s; p; A/, then the marginal values of all

the parameters, including the tertial (non-primal, non-dual) parameter A, exist as

ordinary derivatives. Namely: (i) r s V D r s V D O , (ii) r p V D r p V D yO , and

(iii) r A V D r A V D O ˝ yO D O yO T (the matrix product of a column and a

row, in this order, i.e., the tensor product), where r A is arranged in a matrix like

A (i.e., @V=@Aij D O i yO j for each i and j). The first two formulae (for r s V and

r p V) are cases of a general derivative property of the optimal value in convex

programming: see, e.g., [44, Theorem 16: (b) and (a)] or [32, 7.3: Theorem 1’].

The third formula follows heuristically from either of the first two by comparing

the marginal effect of A with the marginal effect of either s or p on the primal

or dual constraints, respectively. It can also be proved formally by applying the

Generalized Envelope Theorem for smooth optimization [1, (10.8)],8 whereby

each marginal value (r s V, r p V and r A V) is equal to the corresponding partial

derivative of the Lagrangian, which is here

L .y; I p; sI A/ WD

p yC

C1

T

.s

Ay/ if

if

0

.

0

(3.3.1)

• The Kuhn-Tucker Conditions form here the system9

0; Ay Ä s;

T

.Ay

s/ D 0

and pT D

T

A

(3.3.2)

which, because of the quadratic term T Ay in the Complementary Slackness

Condition, is nonlinear in the decision variables (y and jointly).

• By contrast, the FFE Conditions—primal feasibility, dual feasibility and equality

of the primal and dual objectives—form the equivalent system10

Ay Ä s;

0; pT D

T

A

and p y D

s

(3.3.3)

7

The dual constraint AT D p must be changed to AT

p if y 0 is adjoined as another primal

constraint (in which case the primal LP can be interpreted as, e.g., revenue maximization—given

a resource bundle s, an output-price system p and a Leontief technology defined by an inputcoefficient matrix A).

8

Without a proof of value differentiability, the Generalized Envelope Theorem is given also in, e.g.,

[47, 1.F.b].

9

These are the Lagrangian Saddle-Point Conditions (0 2 @ L and 0 2 @O y L) for the present LP

case.

10

In this case, equivalence of the Kuhn-Tucker Conditions and the FFE Conditions can be seen

directly, but it holds always (since, by the general theory of CPs, each system is equivalent to the

conjunction of primal and dual optimality together with absence of a duality gap).

3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price. . .

31

which is linear in .y; /. This makes it simpler to solve than the system of KuhnTucker Conditions (3.3.2). The FFE system (3.3.3) is so effective because, in

linear programming, the dual programme can be worked out from the primal

explicitly.

• But the dual of a general CP cannot be given explicitly (i.e., without leaving

an unevaluated extremum in the formula for the dual constrained objective

function in terms of the Lagrangian).11 That is why, as a general solution method

for convex programming, the Kuhn-Tucker Conditions are better than the FFE

Conditions, although the latter system is simpler in some important specific

cases (such as linear programming). Whereas using the FFE Conditions requires

forming the dual from the primal to start with, using the Kuhn-Tucker Conditions

requires only the Lagrangian. Thus the latter Kuhn-Tucker Conditions offer a

workable general method of solving the primal-dual programme pair, and this

matters more than an explicit expression for the dual programme. The FFE

Conditions can, however, be simpler in the case of a specific CP that, like an

LP, has an explicit dual.

The duality scheme is next applied to all four of the profit and cost programmes

of Sect. 3.1; the one of most importance in the context of a decentralized industry

(such as the ESI of Sects. 5.1 to 5.3) is the programme of SRP maximization. The

duals are shown to consist in shadow-pricing the given quantities—and so their

subprogramme relationship is the reverse of that between the primals: the more

quantities that are fixed, the more commodities there are to shadow-price. (In other

words, the fewer primal variables, the more primal parameters, and hence more

dual variables.) For this reason, the duals are listed, below, in the order reverse to

that in which the primals are listed in Sect. 3.1. See also Fig. 3.1, in which the

large single arrows point from primal programmes to their subprogrammes, and the

double arrows point from the dual programmes to their subprogrammes. Each of

the four middle boxes gives the data for the pair of programmes represented by the

two adjacent boxes (the outer box for the primal and the inner box for the dual); the

data are partitioned into the primal parameters (the given quantities) and the dual

parameters (the given prices). There are no other parameters in this scheme (i.e., it

has no extrinsic parameters).

11

The standard dual to the ordinary CP of maximizing a concave function f .y/ over y subject

to G .y/ Ä s (where G1 , G2 , etc., are convex functions) is to minimize supy L .y; / WD

.s G .y/// over

0 (the standard dual variables, which are the Lagrange

supy . f .y/ C

multipliers for the primal constraints): see, e.g., [44, (5.1)]. And supy L (the Lagrangian’s

supremum over the primal variables) cannot be evaluated without assuming a specific form for

f and G (the primal objective and constraint functions).

32

3 Characterizations of Long-Run Producer Optimum

Fig. 3.1 Decision variables and parameters for primal programmes (optimization of: long-run

profit, short-run profit, long-run cost, short-run cost) and for dual programmes (price consistency

check, optimization of: fixed-input value, output value, output value less fixed-input value). In

each programme pair, the same prices and quantities—. p; y/ for outputs, .r; k/ for fixed inputs,

and .w; v/ for variable inputs—are differently partitioned into decision variables and data (which

are subdivided into primal and dual parameters). Arrows lead from programmes to subprogrammes

In the SRC minimization programme (3.1.10)–(3.1.11), only y and k can serve

as primal parameters,12 and perturbation by both increments, y and k, yields the

following dual programme of shadow-pricing both the outputs and the fixed inputs:

Given .y; k; w/ , maximize h p j yi

hr j ki

…LR . p; r; w/ over . p; r/ 2 P R.

(3.3.4)

Its optimal value is denoted by C SR .y; k; w/ Ä CSR .y; k; w/, with equality when

Sect. 6.2 applies. The dual parameter is w.

In the LRC minimization programme (3.1.8)–(3.1.9), only y can serve as a

primal parameter, and perturbation by the increment y yields the following dual

12

Since the minimand hw j vi is not jointly convex in w and v, w cannot serve as a primal parameter

(it will turn out to be a dual parameter).

3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price. . .

33

Given .y; r; w/ , maximize h p j yi

…LR . p; r; w/ over p 2 P.

(3.3.5)

Its optimal value is denoted by CLR .y; r; w/ Ä CLR .y; r; w/, with equality when

Sect. 6.2 or 6.4 applies. The dual parameters are r and w.

In the SRP maximization programme (3.1.6)–(3.1.7), only k can serve as a

primal parameter, and perturbation by the increment k yields the following dual

programme of shadow-pricing the fixed inputs:

Given . p; k; w/ , minimize hr j ki C …LR . p; r; w/ over r 2 R.

(3.3.6)

Its optimal value is denoted by …SR . p; k; w/ …SR . p; k; w/, with equality when

Sect. 6.2 or 6.4 applies.13 The dual parameters are p and w.

The same programme for r—viz., (3.3.6) or (3.3.13)–(3.3.14) under c.r.t.s.—is

also the dual of the reduced SRP programme in (3.2.2), again with k as the primal

parameter. That is, the reduced and the full primal programmes have the same primal

parameters and the same dual programme. Of course, the two duality relationships

cannot be exactly the same because the two dual parameterizations are different: as

has already been pointed out, the reduced primal programme has fewer variables,

and hence fewer dual parameters, than the full programme, whose data are its primal

and dual parameters. Since both programmes have the same data, it follows that

the reduced one has a datum that is neither a primal nor a dual parameter. In the

case of the reduced SRP programme in (3.2.2), such a datum is w: the only primal

parameter is k, and the only dual parameter is p (since y is the only primal variable).

For comparison, in the full SRP programme (3.1.6)–(3.1.7) both p and w are dual

parameters (paired to the primal variables y and v).14

The LRP maximization programme (3.1.1)–(3.1.2) is, in this context, unusual

because none of its data (p, r, w) can serve as a primal parameter—all of the data

are dual parameters. This means that the intrinsic dual has no decision variable;

formally, it is: Given . p; r; w/, minimize …LR . p; r; w/. Having no variable, the dual

minimand is a constant, and it equals the primal value (…LR ): since the dual is trivial,

there can be no question of a duality gap in this case.

By contrast, the other programme pairs can have duality gaps, especially when

the spaces are infinite-dimensional. But even then a gap can appear only at

an exceptional data point: the primal and dual values are always equal under

Slater’s Condition, as generalized in [44, (8.12)], or the compactness-and-continuity

As the notation indicates, … and C are thought of mainly as dual expressions for … and C

(although duality of programmes is fully symmetric).

14

A similar remark applies to the full and reduced shadow-pricing programmes, (3.3.4) for . p; r/

and the one in (3.4.7) for p alone. Taken as the primal parameterized by w, each has the same dual,

viz., the SRC programme (3.1.10)–(3.1.11). And both of the other vector data (y and k) are dual

parameters for the full programme (3.3.4). But the datum k is neither a dual nor a primal parameter

for the reduced programme in (3.4.7).

13

34

3 Characterizations of Long-Run Producer Optimum

conditions of [44, Example 4’ after (5.13)] and [44, Theorem 18’ (d) or (e)]. In the

problem of profit-maximizing operation of a plant with capacity constraints, Slater’s

Condition requires only that the capacities be strictly positive, i.e., that k

0; in

other words, it is always met unless the plant lacks a component. See Lemma 6.4.1

and Proposition 7.4.2 for details, and Appendix A for a counterexample when k is

not strictly positive.

The partial conjugacy relationships between the dual value functions (CSR , CLR ,

…SR , and …LR D …LR ) can be summarized in a diagram like that in (3.1.12) for the

primal values, but with the arrows reversed (and with bars added to the symbols …

and C):

w

…LR

r

p

.

&

k

y

p

r

w …SR

C LR w

&

.

y

k

CSR

w

.

(3.3.7)

For example, the arrow from the p next to …SR to the y next to C SR indicates that

CSR is, as a function of y, the convex conjugate of …SR as a function of p (with k

and w fixed): i.e., by definition,

˚

CSR .y; k; w/ D sup h p j yi

p

«

…SR . p; k; w/ .

(3.3.8)

Formation of the primal-dual programme pair in a specific case requires formulae

for Y and …LR . When the technology is given by a production set (Y), this requires

working out its support function (…LR ). The task simplifies under c.r.t.s.: …LR is

then ı . j Yı /, the 0-1 indicator of the production cone’s polar (3.1.4). In other

words, Yı is the implicit dual constraint set and, by making the constraint explicit,

the dual programmes can be recast in the same form as the primals. For each primal,

the general form of the dual is specialized to the case of c.r.t.s. in the same way,

viz., by adjoining the constraint . p; r; w/ 2 Yı and deleting the now-vanishing term

…LR from (3.3.4), etc. So the dual programme is to impute optimal values to the

given quantities by pricing them in a way that is consistent with the other, given

prices—i.e., so that the entire price system lies in Yı .

Spelt out, under c.r.t.s. the dual of SRC minimization is the following programme

of maximizing the output value less fixed-input value (OFIV) by shadow-pricing

3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price. . .

35

both the outputs and the fixed inputs:

Given .y; k; w/ , maximize h p j yi

hr j ki over . p; r/

ı

subject to . p; r; w/ 2 Y .

(3.3.9)

(3.3.10)

The dual of LRC minimization is (with c.r.t.s.) the following programme of

maximizing the output value (OV) by shadow-pricing the outputs:

Given .y; r; w/ , maximize h p j yi over p

(3.3.11)

subject to . p; r; w/ 2 Yı .

(3.3.12)

The dual of SRP maximization is (under c.r.t.s.) the following programme of

minimizing the total fixed-input value (FIV) by shadow-pricing the fixed inputs:

Given . p; k; w/ , minimize hr j ki over r

ı

subject to . p; r; w/ 2 Y .

(3.3.13)

(3.3.14)

The dual of LRP maximization has no decision variable, and, with c.r.t.s., it may be

thought of as a price consistency check : its value is 0 if . p; r; w/ 2 Yı , and C1

otherwise. Formally, the dual is:

Given . p; r; w/ , minimize 0 subject to . p; r; w/ 2 Yı .

(3.3.15)

Thus, with c.r.t.s., the dual objectives are “automatic”, and formation of the dual

programmes boils down to working out Yı from a specific cone Y. Two frameworks

for this are provided in Sects. 3.12 and 7.2.

Like the primals, the three duals (of the SRC and LRC and SRP programmes) are

henceforth named after their objectives: OFIV, OV and FIV. Strictly speaking, this

terminology fits only the case of c.r.t.s. for the long run (i.e., the case of a production

cone). But it will be used also when c.r.t.s. are not assumed (in Fig. 3.1, Sect. 3.4

and Tables 3.1 and 3.2).

Comments (Dual of a CP More General Than the Profit and Cost

Programmes)

• The dual can be similarly spelt out for a programme of a more general form, with

a parametric primal maximand

h p j yi

I .y; k/

(3.3.16)

where IW Y K ! R [ fC1g is a bivariate convex function, y is the primal

variable, p and k are the data, of which k is the primal parameter. There is no

explicit constraint, but there is the implicit constraint .y; k/ 2 dom I. The dual

36

3 Characterizations of Long-Run Producer Optimum

Table 3.1 The SRP optimization system with its split form, and four derived differential systems

(three of which are derived directly by applying the DP and FOC, and one indirectly by using also

the SSL)

(3.6.4)–(3.6.5)

.y; v/ 2 @p;w …SR (Type Two)

Dual Part.

Inv. Rule

r 2 @O k …SR (Type One)

m

First-Order Condition

Deriv. Prop. of Opt. Val. (twice)

m Deriv. Prop. of Opt. Val. (twice)

Absorption of No-Gap Cond.

SRP Opt. Sys.

(3.4.1)–(3.4.3)

.y; v/ maxi’es short-run profit

Absorption of No-Gap Cond.

Two-stage

solving

r minimizes fixed-input value

…SR D …SR at . p; k; w/

m

SRC/P Part. Diff. Sys.

(3.5.1)–(3.5.3)

p 2 @y CSR

v 2 @O w CSR

r 2 @O k …SR (Type One)

Split SRP Opt. Sys.

(3.2.2)–(3.2.5)

y maximizes revenue less CSR

v minimizes short-run cost

r minimizes fixed-input value

…SR D …SR at . p; k; w/

Deriv. Prop. of Opt. Val. (twice)

Absorption of No-Gap Cond.

(3.6.6)–(3.6.7)

.y; v/ 2 @p;w …SR (Type One)

r 2 @O k …SR (Type Two)

Subdiff.

Sect. Lem.

O-FIV Part. Diff. Sys.

(3.6.1)–(3.6.3)

y 2 @p …SR

v 2 @O w CSR (Type One)

r 2 @O k …SR

minimand is then

hr j ki C I # . p; r/

(3.3.17)

where I # W Y K ! R [ fC1g is the total (bivariate) convex conjugate of I, r is

the dual variable, and p is the dual parameter. (So the dual and primal parameters

are the coefficients of the primal and dual decision variables, respectively.)

• The profit and cost programmes of Sect. 3.1 are obtained as special cases of

maximizing (3.3.16) when I is equal to the 0-1 indicator of a convex set Y Â Y

K. The conjugate I # is then the support function of Y. If additionally Y is a cone,

then I # is the indicator of the polar Yı , and the programme of minimizing hr j ki

over r subject to . p; r/ 2 Yı is dual to the primal programme of maximizing

h p j yi over y subject to .y; k/ 2 Y (parameterized by k). This is spelt out in the

Proof of Proposition 3.10.1 (where . p; w/ and .y; v/ take place of the above p

and y).

3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price. . .

37

Table 3.2 The SRC optimization system with its split form, and four derived differential systems

(three of which are derived directly by applying the DP and FOC, and one indirectly by using also

the SSL)

(3.6.10)–(3.6.11)

Dual Part.

Inv. Rule

v 2 @O w CSR (Type One)

. p; r/ 2 @y;k CSR (Type Two)

m

First-Order Condition

Deriv. Prop. of Opt. Val. (twice)

m Deriv. Prop. of Opt. Val. (twice)

Absorption of No-Gap Cond.

SRC Opt. Sys.

(3.4.4)–(3.4.6)

Absorption of No-Gap Cond.

Two-stage

solving

v minimizes short-run cost

. p; r/ maxs rev. fix.-inp. val.

CSR D CSR at .y; k; w/

m

O-FIV Part. Diff. Sys.

(3.6.1)–(3.6.3)

y 2 @p …SR

v 2 @O w CSR (Type One)

r 2 @O k …SR

Split SRC Opt. Sys.

(3.4.5)–(3.4.8)

p maximizes revenue less …SR

v minimizes short-run cost

r minimizes fixed-input value

CSR D CSR at .y; k; w/

Deriv. Prop. of Opt. Val. (twice)

Absorption of No-Gap Cond.

(3.6.8)–(3.6.9)

Subdiff.

Sect. Lem.

v 2 @O w CSR (Type Two)

. p; r/ 2 @y;k CSR (Type One)

SRC/P Part. Diff. Sys.

(3.5.1)–(3.5.3)

p 2 @y CSR

v 2 @O w CSR

r 2 @O k …SR (Type One)

• The case of a finite LP, parameterized in the standard way, is obtained when

YD f.y; k/ 2 Rn

˚

Rm W Ay Ä kg , so Yı D . p; r/ 2 Rn

Rm W pDAT r; r

0

«

where A is an m n matrix. With general, possibly infinite-dimensional

˝ spaces,

˛

AW Y ! K is a linear operation, and its adjoint AT W R ! P, defined by AT r j y WD

hr j Ayi, replaces the transposed matrix. In other words, minimization of hr j ki

over r subject to p D AT r and r

0 is dual to maximization of h p j yi over y

subject to Ay Ä k (with k as the primal parameter vector).

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3 Duality: Cost and Profit as Values of Programmes with Shadow-Price Decisions

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