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# Complementary slackness and farkas lemmaa

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to test the optimality of dual solutions: complementary slackness theorem
to test the feasibility : farkas' lemma

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### Complementary slackness and farkas lemmaa

1. 1. Complementary slackness & Farkas’ lemma<br />SHAMEER P H<br />DEPT OF FUTURES STUDIES<br />
2. 2. STRONG DUALITY THEOREM<br />If ZLP or WLP is finite, then both P and D have finite optimal value and ZLP= WLP<br />COROLLARY:<br />There are only four possibilities for a dual pair of problems P and D<br />ZLP or WLP are finite and equal<br />ZLP= ∞ and D is infeasible<br />WLP = -∞ and P is infeasible<br />both P and D are infeasible<br />DEPT OF FUTURES STUDIES<br />
3. 3. Complementary Slackness <br />PRIMAL: u = max{cx : Ax≤ b, x≥ 0}<br />DUAL: w = min{by: Ay≥c, y ≥0}<br />let‘s’ be the vector of slackvariables of theprimal&‘t’ isthe vector of surplus variables of dual<br />s= b-Ax≥0 and t= yA-c≥0<br />DEPT OF FUTURES STUDIES<br />
4. 4. Principle<br />If x* isanoptimalsolution of Primal (P) and y *isanoptimalsolution of Dual (D), then<br />xj* tj * =0for all j &<br />yi* si * =0 foralli<br />(The theorem identifies a relationship between variables in one problem and associated constraints in the other problem.)<br />DEPT OF FUTURES STUDIES<br />
5. 5. proof<br />Using the definitions of s and t <br /> cx* = (y*A-t*)x*<br />= y*Ax*- t*x*<br />= y*(b-s*)-t*x* (since Ax*+s=b)<br />= y*b-y*s*-t*x*<br />but by strong duality theorem, cx*=y*b<br />ie; <br />cx* = cx*-(y*s*+t*x*)<br />0= y*s*+t*x*<br />y*s*=0 and t*x*=0 (Since, y*,x*,s*, t*≥0)<br />DEPT OF FUTURES STUDIES<br />
6. 6. In other words<br />Given a dual pair of LPPs have optimal solutions, then if the kth constraint of one system holds inequality (i.e, associated slack or surplus variable is positive) then the kthcomponent of the optimal solution of its dual is zero<br />OR<br /><ul><li>If a variable is positive, then its dual constraint is tight.
7. 7. If a constraint is loose, then its dual variable is zero.</li></ul>DEPT OF FUTURES STUDIES<br />
8. 8. example<br />PRIMAL:<br />max z=6x1 +5x2<br />such that:<br /> x1+x2≤5<br /> 3x1+2x2≤12<br /> x1,x2 ≥0<br />OR<br /> x1+x2+S1=5<br /> 3x1+2x2+S2=12<br /> x1,x2,s1,s2 ≥0<br />Solution:<br /> x1 =2, x2 =3, z=27<br /> S1=0, s2=0<br />DUAL:<br />Min. W=5y1+12y2<br />Such that <br /> y1+3y2≥6<br />y1+2y2≥5<br />y1,y2≥0<br />OR<br /> y1+3y2-t1 =6<br /> y1+2y2-t2=5<br /> y1,y2,t1,t2≥0<br />Solution:<br /> y1 =3, y2 =1, w=27<br />t1 =0, t2=0<br />DEPT OF FUTURES STUDIES<br />
9. 9. applications<br /><ul><li>Used in finding an optimal primal solution from the given optimal dual solution and vice versa
10. 10. Used in finding a feasible solution is optimal for the primal problem </li></ul>DEPT OF FUTURES STUDIES<br />
11. 11. FARKAS’ LEMMA<br />Proposed by Farkas in 1902.<br />Used to check whether a system of linear inequalities is feasible or not <br />Let A be an m × n matrix, b ∈ Rn+. Then exact one of the below is true:<br />There exists an x∈ Rn+suchthat Ax≤ b; or<br />There exists a y∈ Rm+such that y ≥ 0, yA = 0 and yb < 0.<br /> OR<br />Ax≤ b, x ≥ 0 is infeasibleiff yA ≥0, y b < 0 is feasible<br />DEPT OF FUTURES STUDIES<br />
12. 12. Proof:<br />Consider the LPP, ZLP =max{0x:Ax≤b, xԐRn+}<br /> and its dual WLP =min{yb:yA≥0,yԐRm+} .<br />As y=0 is a feasible solution to the dual problem, the only possibilities to occur are i & iii of corollary of strong dual theorem . <br />ZLP =WLP =0. hence {xԐRn+ :Ax ≤b}≠ф and Yb≥0 for all yԐRm+ with yA ≥0<br />WLP = -∞. hence {xԐRn+ :Ax ≤b}=ф and there exists yԐRm+ with yA ≥0 and yb˂0.<br />DEPT OF FUTURES STUDIES<br />
13. 13. Variants of Farkas’ Lemma<br />There are many variants of Farkas’ Lemma.<br />Some are:<br />Either {xԐRn+ :Ax=b}≠ф, or {yԐRm:yA ≥0 ,yb˂0} ≠ф<br />Either {xԐRn:Ax≤b}≠ф, or {yԐRm+ : yA=0 , yb˂0} ≠ф<br />DEPT OF FUTURES STUDIES<br />
14. 14. THANK YOU….<br />DEPT OF FUTURES STUDIES<br />