5 The Primal-Dual Simplex Algorithm

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1 he Primal-Dual Simplex Algorithm Again, we consider the primal program given as a minimization problem defined in standard form his algorithm is based on the cognition that both optimal solutions, i.e., the primal and the dual one, are strongly interdependent Specifically, the approach commences the searching process with a feasible dual solution and simultaneously observes the complementary slackness between the solution value of the dual and a primal solution If this slackness becomes zero, the optimality of the generated solutions is proven and the calculation process is terminated Business Computing and Operations Research 46

2 Invariants of the Primal Simplex While conducting the Primal Simplex, the following attributes are always fulfilled for a minimization problem: P Minimize c x, s.t. A x = b x B B N N B B. c x = c x + c x = c A b = π b = b π. c x = c x + c x = x + c =, with B B B B c = c c A A N N B N B B hus, if c c A A = π A c π is feasible for D Maximize b π, s.t. A π c π free Business Computing and Operations Research 4

3 Consequences he Primal Simplex works on a feasible primal solution that is iteratively improved by basis changes his is done by the consideration of a corresponding dual solution that has an identical objective function value As long as this dual solution is infeasible, the corresponding entries are inserted in the primal solution in order to fulfill them exactly in the dual program (Elimination of the corresponding slackness) If the dual solution becomes feasible as well the optimality of both solutions (the primal and the dual solution) is proven Business Computing and Operations Research 4

4 he Primal-Dual Simplex As mentioned above, we assume that the primal program is given as a minimization problem in standard form In what follows, we introduce a new algorithm that commences the searching process with a feasible dual solution his solution is analyzed according to a specific relationship to the primal problem in order to generate a corresponding primal solution that allows to prove optimality Specifically, we formulate a reduced problem that either generates an optimal primal solution or, if this is not possible, allows a correction of the dual one Obviously, this process is executed until the first case applies Business Computing and Operations Research 49

5 Observation. heorem of Complementary Slackness Assuming there is a Linear Program in standard form and x and π are feasible solutions to P and D, respectively. hen, it holds: ( ) x and π are optimal c π A x = Business Computing and Operations Research 46

6 Proof of heorem. Fortunately, this proof is quite easy to conduct Based on the facts we already know about tuples of optimal primal and dual solutions, we derive Specifically, it holds: c π A x = c x π A x = Since x is feasible for c x π b = c x = π b x P and π are optimal solutions Business Computing and Operations Research 46

7 Direct consequence of heorem.. Observation ( ) Assuming x and π are feasible solutions to P and D, respectively. Additionally, assume that it holds: c π A x =. hus, x and π are optimal solutions to P and D, respectively and it holds: c π A x Hence, we can conclude ( j c ) j π a x j =, j {,..., n} { } j x = π a = c, j,..., n j j Business Computing and Operations Research 46

8 A simple example Consider the following Linear Program P Minimize, 3,, x s.t. x = x 3 9 Business Computing and Operations Research 463

9 Example hus, we get the following (D) D Maximize, 9 π 3 s.t. π π free 3 Business Computing and Operations Research 464

10 Example How to generate π? Obviously, x = is a feasible solution to ( P). 6 hus, we have x = x = x, x. Consequently, we 3 4 need a π with the following attributes i. i { 3, 4} :π a = ci π = 3. π is feasible for D Business Computing and Operations Research 46

11 Example How to generate π? Obviously, π = fulfills both restrictions and, therefore, we have shown that x = and π = are optimal solutions to ( P) and ( D), 6 respectively. Business Computing and Operations Research 466

12 A feasible solution to (D) For what follows, we need at first a feasible solution to the dual problem. Fortunately, this is quite simple to provide. If c is positive, we just make use of π=. Otherwise, we apply the following simple procedure that is depicted next. Business Computing and Operations Research 46

13 Generating a feasible dual solution In order to generate a feasible dual solution to cases where c does not apply, we provide the following simple construction procedure. We introduce a n + th variable x as well as a m + th equality in n+ x + x x + x = x = b, with b as a huge n+ n n+ i m+ m+ i= number. Since we add impact (P) c n+ = on the optimal solutions., we know that this restriction has no Business Computing and Operations Research 46

14 Generating a feasible dual solution. Consider now the dual program. Maximize b π + bm + πm+ A π π m+... πm+ c πm+ + j j : π a + π c m+ j, i.e., ( π ) 3. We generate π = π,..., as follows: ini ini ini m+ { } π = π =... = π = π = min c c < < ini ini ini m m+ j j { } ( D) hus, since j,..., n exists with c <, π ini is feasible for. Business Computing and Operations Research 469 j

15 he set J Assume π to be a feasible solution to the dual program of a Linear Program in standard form. An index j,...,n is denoted as feasible if and only if it holds: { } j π a = c. j We introduce J as the set of feasible indices, i.e., { { } j } J = j j,...,n π a = c. j Business Computing and Operations Research 4

16 We assume J Reduced primal problem (RP) J = j,..., j,k { } ( j ) jk and a J A = a,...,a x = x,x, ( x ),...,xm and define a a a with x = as slack variables. k hen RP is defined as follows: a a a x x Minimize ξ = x, s.t. E m, AJ b, with J = J x x is denoted as the reduced primal problem. Business Computing and Operations Research 4

17 Observations (RP) is solvable. Specifically, we can use x =(b,) Since this trivial solution has the objective function value.b and this objective function is lower bounded by, (RP) is bounded hus, (RP) has always a well-defined optimal solution Obviously, this optimal solution comprises two parts First, there are the slackness variables. If these are zero, the objective function value is zero as well. hen, the primal solution is optimal to (P) Secondly, there are the original variables that correspond to the set J. Since the corresponding dual values are equal to the c-vector, only these variables may become unequal to zero Business Computing and Operations Research 4

18 .3 heorem ( RP) Main conclusion a always has an optimal solution. If x = ξ =, is an optimal solution to. J P x Otherwise, if ξ >, then the optimal solution to the dual ( RP) ( D) of always generates an improved solution to. Business Computing and Operations Research 43

19 Proof of heorem.3 Case a At first, we assume ξ =. hus, we know that x =. Consequently, J J it holds: A x = b. hus, we consider xˆ = A xˆ b J = ( c π A) x x ( c π A) xj ( c π A) x c c xj ( c π A) = + = + = J J J J Hence, x and π are optimal solutions to the Linear Programs P and D, respectively. c Business Computing and Operations Research 44

20 Proof of heorem.3 Case a Now, we consider the case ξ >. hus, we know that x. Consequently, it holds: J J A x b. Let us now consider the dual of ( RP) ( RP), denoted as ( DRP) a a Minimize s.t. ( J x ) ( a J x, E ) m, A = b,x = x,x J x hus, ( DRP) we obtain as follows E m m Maximize b π, s.t. free ( J ) π,π J A Business Computing and Operations Research 4

21 Proof of heorem.3 Case Assuming πɶ is an optimal solution to. hen, we conclude b πɶ = ξ >. Furthermore, let π = π + λ πɶ. We compute b π b π λ π b π b λ π b π + λ b πɶ = b π + λ ξ > b π. DRP = + ɶ = + ɶ = Consequently, if π is feasible for D, π outperforms π. Hence, we now have to determine suitable values for λ resulting in feasible values for π. Business Computing and Operations Research 46

22 Proof of heorem.3 Case Note that it holds: π feasible for j j : c j π a j j : c j π + λ πɶ a j j j : c j π a λ π a j : c π a λ πɶ a j ɶ j j D Business Computing and Operations Research 4

23 j Proof of heorem.3 Case j : c π a λ π ɶ a j j j Since π is feasible for D, we know that c π a. Let us now consider the corresponding final tableau of ( RP) ξ π ɶ π ɶ A π ɶ A c J J b E A A J J m j c J J j π ɶ A π ɶ A j J :π ɶ a Hence, if fulfilled. j J λ >, the feasibility restriction is always Business Computing and Operations Research 4

24 Proof of heorem.3 Case ξ π ɶ π ɶ A π ɶ A c J J b E A A J J m c c J However, since A does not belong to RP, there may be j J j with π ɶ a >. If so, we can determine j c j π a c j λ = min j J :π ɶ a > j. π ɶ a ( P) { } Consequently, π + λ π ɶ is feasible for λ,..., λ. j If there is, however, no j with π ɶ a >, D is unbounded and, consequently, is not solvable. Business Computing and Operations Research 49 c

25 Summary he algorithm. We commence the searching process with a feasible solution π to the dual program (D).. hen, we generate the reduced Linear Program (RP(π)) and solve it optimally. hus, we distinguish altogether three cases:. ξ = he tableau provides an optimal solution to P. Business Computing and Operations Research 4

26 ( P) Summary he algorithm Further cases (Continuation of step ) j. ξ > j :π ɶ a he primal problem is not solvable. j 3. ξ > j :π ɶ a > Generate π = π + λ π,π ɶ ɶ optimal solution to the problem DRP π. j c j π a c j Determine λ = min j J :π ɶ a > j. π ɶ a Repeat step until one of the cases or applies. Business Computing and Operations Research 4

27 Illustration (P) (D) π feasible (RP(π)) (DRP(π)) ξ = () Update π π + λ π~ Solutions : x andπ: ξ > No j exists () () () x and π are optimal. ermination Since ( P) ( ɶ) b π + λ π, is not solvable. ermination Business Computing and Operations Research 4

28 he Primal-Dual Simplex Algorithm. ransform the problem such that b and generate equations.. Initialization with a feasible basic solution to the dual problem. 3. Determine the set J = j =,...,n π a = c. 4. Solve the reduced primal problem (RP) to optimality via the Primal Simplex Algorithm:. If ξ =, then the optimal solution to the primal problem (P) is found. erminate and calculate the objective function value Z with the basic variables of J: Otherwise (i.e., ξ > ) : 6. Calculate the dual variables πɶ with cost coefficients of RP belonging to x a : π ɶ = m c j. If j ξ > j :π ɶ a, then terminate since the primal problem (P) is unbounded and no optimal solution exists. j Otherwise (i.e., ξ ) : > j : π ɶ a > j c j π a j. Determine λ = min j J :π ɶ a >. j π ɶ a 9. Update the dual variables: π : = π + λ ɶπ.. Go to step 3. j { j} m a a J J a J RP ξ = Min x s.t. Em x + A x = b x,x J j Z = c x j = Business Computing and Operations Research 43 j J,...,m

29 Example ( P) Minimize (,3 ) x,s.t., 3 4 x x ( P ) s Minimize ( 3 ) x,s.t., 3 4 x = x Business Computing and Operations Research 44

30 Example ( D) Maximize (,,, ) π,s.t., - π π free - - Since c, we can apply the trivial dual solution π =. { } J { } c hus, we get J =, = 3, 4, Business Computing and Operations Research 4

31 ( RP ( π) ) Example Generate (RP(π)) Minimize ( ) x, s.t., x x = In order, however, to identify the j - values, we integrate the columns of c set J in the tableau as well. hus, we obtain Business Computing and Operations Research 46

32 Example Generate λ 3 4 ( ) ( ) πɶ πɶ ( ) = = π a 3 π a 3 λ = min, min =, ( ) a ( ) a = min, = Business Computing and Operations Research 4

33 Example Generate λ Consequently, we obtain for the next round π = ( ) + ( ) = hus, since ξ = >, we get a new ( RP ( π) ) At first, we have to identify J. Business Computing and Operations Research 4

34 Business Computing and Operations Research 49 Example Generate J { } { },3,4, time we obtain this hus, 3 / / / / / / / / 4 / 4 3 we determine herefore, = = = = = = c J J c π

35 Business Computing and Operations Research 49 Example Solving (RP(π)) [ ]

36 Example Solving (RP(π)) ( ) ~ ~ ( ),, =,, π π =,, λ 3 = min = min, = 9 3 9, ( ) = min 3, Business Computing and Operations Research 49

37 Example Updating π and J Consequently, we obtain for the next round ( ) π = + = hus, since ξ = 9 >, we get a new ( RP( π) ) At first, we again have to identify J. π / / = = 4 / 3 = 4 / 3 = / 3 4 / / 3 3 / 3 c hus, this time we obtain J, J c = { } = { 3, 4, } Business Computing and Operations Research 49

38 Business Computing and Operations Research 493 Example Solving (RP(π)) [ ] ~ ~ = =,, π π,,,,

39 Example Solving (RP(π)) ( 3 ) ~ ~ ( ),, =,, π π =,, λ 4 3 min 3, 3 = = hus, we can updateπ = = = ( ) 4 ( ),, +,, ( ), +, 3 ( 4 ) ( + 4) ( 6 6) ( 4 ) ( 4 ),, =,,, 9 by : π = 9 3, Business Computing and Operations Research 494

40 Business Computing and Operations Research 49 Example Updating J { } { } 3,4,, we obtain time this hus, 3 4 / / 3 4 / / / 4 / we have to identify Again, = = = = = = c J J c π J

41 Example Solving (RP(π)) ( ) ξ = Optimal solutions are: ( 4 ) ( 6 ) π =,, x = Business Computing and Operations Research 496

42 Additional literature to Section he primal-dual algorithm for general LP's was first described in Dantzig, G.B.: Ford, L.R.; Fulkerson, D.R. (96): A Primal-Dual Algorithm for Linear Programs," in Kuhn, H.W.; ucker, A.W. (eds.): Linear Inequalities and Related Systems. Princeton University Press, Princeton, N.J., pp. -. It is introduced there as a generalization of the paper Kuhn, H.W. (9): he Hungarian Method for the Assignment Problem. Naval Research Logistics Quarterly,, nos. and, pp Business Computing and Operations Research 49

Dual Basic Solutions. Observation 5.7. Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP:

Dual Basic Solutions. Observation 5.7. Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP: Dual Basic Solutions Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP: Observation 5.7. AbasisB yields min c T x max p T b s.t. A x = b s.t. p T A apple c T x 0 aprimalbasicsolutiongivenbyx