Interpretation 1: Economic Interpretation Primal LP � n max j =1 c j x j � n s.t. j =1 a ij x j ≤ b i , for i ∈ [ m ] . x j ≥ 0 , for j ∈ [ n ] . Duality and Its Interpretations 14/40
Interpretation 1: Economic Interpretation Primal LP Dual LP � n � m max j =1 c j x j min i =1 b i y i � n � m s.t. j =1 a ij x j ≤ b i , for i ∈ [ m ] . s.t. i =1 a ij y i ≥ c j , for j ∈ [ n ] . x j ≥ 0 , for j ∈ [ n ] . y i ≥ 0 , for i ∈ [ m ] . Duality and Its Interpretations 14/40
Interpretation 1: Economic Interpretation Primal LP Dual LP � n � m max j =1 c j x j min i =1 b i y i � n � m s.t. j =1 a ij x j ≤ b i , for i ∈ [ m ] . s.t. i =1 a ij y i ≥ c j , for j ∈ [ n ] . x j ≥ 0 , for j ∈ [ n ] . y i ≥ 0 , for i ∈ [ m ] . x 1 x 2 x 3 x 4 y 1 a 11 a 12 a 13 a 14 b 1 y 2 a 21 a 22 a 23 a 24 b 2 y 3 a 31 a 32 a 33 a 34 b 3 c 1 c 2 c 3 c 4 Duality and Its Interpretations 14/40
Interpretation 1: Economic Interpretation Primal LP Dual LP � n � m max j =1 c j x j min i =1 b i y i � n � m s.t. j =1 a ij x j ≤ b i , for i ∈ [ m ] . s.t. i =1 a ij y i ≥ c j , for j ∈ [ n ] . x j ≥ 0 , for j ∈ [ n ] . y i ≥ 0 , for i ∈ [ m ] . x 1 x 2 x 3 x 4 y 1 a 11 a 12 a 13 a 14 b 1 y 2 a 21 a 22 a 23 a 24 b 2 y 3 a 31 a 32 a 33 a 34 b 3 c 1 c 2 c 3 c 4 Dual variable y i is a proposed price per unit of raw material i Dual price vector is feasible if facility has incentive to sell materials Buyer wants to spend as little as possible to buy materials Duality and Its Interpretations 14/40
Interpretation 2: Finding the Best Upperbound Consider the simple LP from last lecture maximize x 1 + x 2 subject to x 1 + 2 x 2 ≤ 2 2 x 1 + x 2 ≤ 2 x 1 , x 2 ≥ 0 We found that the optimal solution was at ( 2 3 , 2 3 ) , with an optimal value of 4 / 3 . Duality and Its Interpretations 15/40
Interpretation 2: Finding the Best Upperbound Consider the simple LP from last lecture maximize x 1 + x 2 subject to x 1 + 2 x 2 ≤ 2 2 x 1 + x 2 ≤ 2 x 1 , x 2 ≥ 0 We found that the optimal solution was at ( 2 3 , 2 3 ) , with an optimal value of 4 / 3 . What if, instead of finding the optimal solution, we saught to find an upperbound on its value by combining inequalities? Each inequality implies an upper bound of 2 Multiplying each by 1 3 and summing gives x 1 + x 2 ≤ 4 / 3 . Duality and Its Interpretations 15/40
Interpretation 2: Finding the Best Upperbound x 1 x 2 x 3 x 4 y 1 a 11 a 12 a 13 a 14 b 1 y 2 a 21 a 22 a 23 a 24 b 2 y 3 a 31 a 32 a 33 a 34 b 3 c 1 c 2 c 3 c 4 Multiplying each row i by y i and summing gives the inequality y T Ax ≤ y T b Duality and Its Interpretations 16/40
Interpretation 2: Finding the Best Upperbound x 1 x 2 x 3 x 4 y 1 a 11 a 12 a 13 a 14 b 1 y 2 a 21 a 22 a 23 a 24 b 2 y 3 a 31 a 32 a 33 a 34 b 3 c 1 c 2 c 3 c 4 Multiplying each row i by y i and summing gives the inequality y T Ax ≤ y T b When y T A ≥ c T , the right hand side of the inequality is an upper bound on c T x for every feasible x . c T x ≤ y T Ax ≤ y T b Duality and Its Interpretations 16/40
Interpretation 2: Finding the Best Upperbound x 1 x 2 x 3 x 4 y 1 a 11 a 12 a 13 a 14 b 1 y 2 a 21 a 22 a 23 a 24 b 2 y 3 a 31 a 32 a 33 a 34 b 3 c 1 c 2 c 3 c 4 Multiplying each row i by y i and summing gives the inequality y T Ax ≤ y T b When y T A ≥ c T , the right hand side of the inequality is an upper bound on c T x for every feasible x . c T x ≤ y T Ax ≤ y T b The dual LP can be thought of as trying to find the best upperbound on the primal that can be achieved this way. Duality and Its Interpretations 16/40
Interpretation 3: Physical Forces Apply force field c to a ball inside bounded polytope Ax � b . Duality and Its Interpretations 17/40
Interpretation 3: Physical Forces Apply force field c to a ball inside bounded polytope Ax � b . Eventually, ball will come to rest against the walls of the polytope. Duality and Its Interpretations 17/40
Interpretation 3: Physical Forces Apply force field c to a ball inside bounded polytope Ax � b . Eventually, ball will come to rest against the walls of the polytope. Wall a i x ≤ b i applies some force − y i a i to the ball Duality and Its Interpretations 17/40
Interpretation 3: Physical Forces Apply force field c to a ball inside bounded polytope Ax � b . Eventually, ball will come to rest against the walls of the polytope. Wall a i x ≤ b i applies some force − y i a i to the ball Since the ball is still, c T = � i y i a i = y T A . Duality and Its Interpretations 17/40
Interpretation 3: Physical Forces Apply force field c to a ball inside bounded polytope Ax � b . Eventually, ball will come to rest against the walls of the polytope. Wall a i x ≤ b i applies some force − y i a i to the ball Since the ball is still, c T = � i y i a i = y T A . Dual can be thought of as trying to minimize “work” � i y i b i to bring ball back to origin by moving polytope We will see that, at optimality, only the walls adjacent to the ball push (Complementary Slackness) Duality and Its Interpretations 17/40
Outline Linear Programming Basics 1 Duality and Its Interpretations 2 3 Properties of Duals Weak and Strong Duality 4 5 Formal Proof of Strong Duality of LP Consequences of Duality 6 More Examples of Duality 7
Duality is an Inversion Primal LP Dual LP maximize c ⊺ x minimize b ⊺ y subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Duality is an Inversion Given a primal LP in standard form, the dual of its dual is itself. Properties of Duals 18/40
Correspondance Between Variables and Constraints Primal LP Dual LP � n � m max j =1 c j x j min i =1 b i y i s.t. s.t. � n � m j =1 a ij x j ≤ b i , for i ∈ [ m ] . i =1 a ij y i ≥ c j , for j ∈ [ n ] . y i ≥ 0 , for i ∈ [ m ] . x j ≥ 0 , for j ∈ [ n ] . Properties of Duals 19/40
Correspondance Between Variables and Constraints Primal LP Dual LP � n � m max j =1 c j x j min i =1 b i y i s.t. s.t. � n � m y i : j =1 a ij x j ≤ b i , for i ∈ [ m ] . i =1 a ij y i ≥ c j , for j ∈ [ n ] . y i ≥ 0 , for i ∈ [ m ] . x j ≥ 0 , for j ∈ [ n ] . The i ’th primal constraint gives rise to the i ’th dual variable y i Properties of Duals 19/40
Correspondance Between Variables and Constraints Primal LP Dual LP � n � m max j =1 c j x j min i =1 b i y i s.t. s.t. � n � m y i : j =1 a ij x j ≤ b i , for i ∈ [ m ] . x j : i =1 a ij y i ≥ c j , for j ∈ [ n ] . y i ≥ 0 , for i ∈ [ m ] . x j ≥ 0 , for j ∈ [ n ] . The i ’th primal constraint gives rise to the i ’th dual variable y i The j ’th primal variable x j gives rise to the j ’th dual constraint Properties of Duals 19/40
Syntactic Rules Primal LP Dual LP min b ⊺ y max c ⊺ x s.t. s.t. x j : a ⊺ j y ≥ c j , for j ∈ D 1 . y i : a i x ≤ b i , for i ∈ C 1 . a ⊺ y i : a i x = b i , for i ∈ C 2 . x j : j y = c j , for j ∈ D 2 . x j ≥ 0 , for j ∈ D 1 . y i ≥ 0 , for i ∈ C 1 . x j ∈ R , for j ∈ D 2 . y i ∈ R , for i ∈ C 2 . Rules of Thumb Lenient constraint (i.e. inequality) ⇒ stringent dual variable (i.e. nonnegative) Stringent constraint (i.e. equality) ⇒ lenient dual variable (i.e. unconstrained) Properties of Duals 20/40
Outline Linear Programming Basics 1 Duality and Its Interpretations 2 3 Properties of Duals Weak and Strong Duality 4 5 Formal Proof of Strong Duality of LP Consequences of Duality 6 More Examples of Duality 7
Weak Duality Primal LP Dual LP maximize c ⊺ x minimize b ⊺ y subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Theorem (Weak Duality) For every primal feasible x and dual feasible y , we have c ⊺ x ≤ b ⊺ y . Corollary If primal and dual both feasible and bounded, OPT ( Primal ) ≤ OPT ( Dual ) If primal is unbounded, dual is infeasible If dual is unbounded, primal is infeasible Weak and Strong Duality 21/40
Weak Duality Primal LP Dual LP maximize c ⊺ x minimize b ⊺ y subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Theorem (Weak Duality) For every primal feasible x and dual feasible y , we have c ⊺ x ≤ b ⊺ y . Corollary If x ∗ is primal feasible, and y ∗ is dual feasible, and c ⊺ x ∗ = b ⊺ y ∗ , then both are optimal. Weak and Strong Duality 21/40
Interpretation of Weak Duality Economic Interpretation If selling the raw materials is more profitable than making any individual product, then total money collected from sale of raw materials would exceed profit from production. Weak and Strong Duality 22/40
Interpretation of Weak Duality Economic Interpretation If selling the raw materials is more profitable than making any individual product, then total money collected from sale of raw materials would exceed profit from production. Upperbound Interpretation Self explanatory Weak and Strong Duality 22/40
Interpretation of Weak Duality Economic Interpretation If selling the raw materials is more profitable than making any individual product, then total money collected from sale of raw materials would exceed profit from production. Upperbound Interpretation Self explanatory Physical Interpretation Work required to bring ball back to origin by pulling polytope is at least potential energy difference between origin and primal optimum. Weak and Strong Duality 22/40
Proof of Weak Duality Primal LP Dual LP maximize c ⊺ x minimize b ⊺ y subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 c ⊺ x ≤ y ⊺ Ax ≤ y ⊺ b Weak and Strong Duality 23/40
Strong Duality Primal LP Dual LP maximize c ⊺ x minimize b ⊺ y subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Theorem (Strong Duality) If either the primal or dual is feasible and bounded, then so is the other and OPT ( Primal ) = OPT ( Dual ) . Weak and Strong Duality 24/40
Interpretation of Strong Duality Economic Interpretation Buyer can offer prices for raw materials that would make facility indifferent between production and sale. Weak and Strong Duality 25/40
Interpretation of Strong Duality Economic Interpretation Buyer can offer prices for raw materials that would make facility indifferent between production and sale. Upperbound Interpretation The method of scaling and summing inequalities yields a tight upperbound on the primal optimal value. Weak and Strong Duality 25/40
Interpretation of Strong Duality Economic Interpretation Buyer can offer prices for raw materials that would make facility indifferent between production and sale. Upperbound Interpretation The method of scaling and summing inequalities yields a tight upperbound on the primal optimal value. Physical Interpretation There is an assignment of forces to the walls of the polytope that brings ball back to the origin without wasting energy. Weak and Strong Duality 25/40
Informal Proof of Strong Duality Recall the physical interpretation of duality Weak and Strong Duality 26/40
Informal Proof of Strong Duality Recall the physical interpretation of duality When ball is stationary at x , we expect force c to be neutralized only by constraints that are tight. i.e. force multipliers y � 0 s.t. y ⊺ A = c y i ( b i − a i x ) = 0 Weak and Strong Duality 26/40
Informal Proof of Strong Duality Recall the physical interpretation of duality When ball is stationary at x , we expect force c to be neutralized only by constraints that are tight. i.e. force multipliers y � 0 s.t. y ⊺ A = c y i ( b i − a i x ) = 0 � y ⊺ b − c ⊺ x = y ⊺ b − y ⊺ Ax = y i ( b i − a i x ) = 0 i We found a primal and dual solution that are equal in value! Weak and Strong Duality 26/40
Outline Linear Programming Basics 1 Duality and Its Interpretations 2 3 Properties of Duals Weak and Strong Duality 4 5 Formal Proof of Strong Duality of LP Consequences of Duality 6 More Examples of Duality 7
Separating Hyperplane Theorem If A, B ⊆ R n are disjoint convex sets, then there is a hyperplane separating them. That is, there is a ∈ R n and b ∈ R such that a ⊺ x ≤ b for every x ∈ A and a ⊺ y ≥ b for every y ∈ B . Moreover, if both A and B are closed and at least one of them is compact, then there is a hyperplane strictly separating them (i.e. a T x < b for x ∈ A and a T y > b for y ∈ B ). Formal Proof of Strong Duality of LP 27/40
Definition A convex cone is a convex subset of R n which is closed under nonnegative scaling and convex combinations. Definition The convex cone generated by vectors u 1 , . . . , u m ∈ R n is the set of all nonnegative-weighted sums of these vectors (also known as conic combinations). � m � � Cone ( u 1 , . . . , u m ) = α i u i : α i ≥ 0 ∀ i i =1 Formal Proof of Strong Duality of LP 28/40
The following follows from the separating hyperplane Theorem (try to prove it). Farkas’ Lemma Let C be the convex cone generated by vectors u 1 , . . . , u m ∈ R n , and let w ∈ R n . Exactly one of the following is true: w ∈ C There is z ∈ R n such that z · u i ≤ 0 for all i , and z · w > 0 . Formal Proof of Strong Duality of LP 28/40
Equivalently: Theorem of the Alternative Exactly one of the following is true for U = [ u 1 , . . . , u m ] and w The system Uz = w , z � 0 has a solution The system U ⊺ z � 0 , z ⊺ w > 0 has a solution. Formal Proof of Strong Duality of LP 28/40
Formal Proof of Strong Duality Primal LP Dual LP maximize c ⊺ x minimize b ⊺ y subject to Ax � b subject to A ⊺ y = c y � 0 Given v ∈ R , by Farkas’ Lemma exactly one of the following is true � A ⊺ 0 � � c � The system z = , z � 0 has a solution. 1 b ⊺ 1 v � y � Let y ∈ R m + and δ ∈ R + be such that z = δ Implies dual is feasible and OPT ( dual ) ≤ v Formal Proof of Strong Duality of LP 29/40
Formal Proof of Strong Duality Primal LP Dual LP maximize c ⊺ x minimize b ⊺ y subject to Ax � b subject to A ⊺ y = c y � 0 Given v ∈ R , by Farkas’ Lemma exactly one of the following is true � A ⊺ 0 � � c � The system z = , z � 0 has a solution. 1 b ⊺ 1 v � y � Let y ∈ R m + and δ ∈ R + be such that z = δ Implies dual is feasible and OPT ( dual ) ≤ v � A b � � c � The system z � 0 , z ⊺ > 0 has a solution. 2 0 1 v � z 1 � , where z 1 ∈ R n and z 2 ∈ R with z 2 ≤ 0 Let z = z 2 When z 2 � = 0 , x = − z 1 /z 2 is primal feasible and c T x > v A Formal Proof of Strong Duality of LP 29/40
Formal Proof of Strong Duality Primal LP Dual LP maximize c ⊺ x minimize b ⊺ y subject to Ax � b subject to A ⊺ y = c y � 0 Given v ∈ R , by Farkas’ Lemma exactly one of the following is true � A ⊺ 0 � � c � The system z = , z � 0 has a solution. 1 b ⊺ 1 v � y � Let y ∈ R m + and δ ∈ R + be such that z = δ Implies dual is feasible and OPT ( dual ) ≤ v � A b � � c � The system z � 0 , z ⊺ > 0 has a solution. 2 0 1 v � z 1 � , where z 1 ∈ R n and z 2 ∈ R with z 2 ≤ 0 Let z = z 2 When z 2 � = 0 , x = − z 1 /z 2 is primal feasible and c T x > v A When z 2 = 0 , primal is either infeasible or unbounded, and dual is B infeasible (prove it) Formal Proof of Strong Duality of LP 29/40
Outline Linear Programming Basics 1 Duality and Its Interpretations 2 3 Properties of Duals Weak and Strong Duality 4 5 Formal Proof of Strong Duality of LP Consequences of Duality 6 More Examples of Duality 7
Complementary Slackness Primal LP Dual LP maximize c ⊺ x minimize y ⊺ b subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Consequences of Duality 30/40
Complementary Slackness Primal LP Dual LP maximize c ⊺ x minimize y ⊺ b subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Let s i = ( b − Ax ) i be the i ’th primal slack variable Let t j = ( A ⊺ y − c ) j be the j ’th dual slack variable Consequences of Duality 30/40
Complementary Slackness Primal LP Dual LP maximize c ⊺ x minimize y ⊺ b subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Let s i = ( b − Ax ) i be the i ’th primal slack variable Let t j = ( A ⊺ y − c ) j be the j ’th dual slack variable Complementary Slackness x 1 x 2 x 3 x 4 Feasible x and y are optimal if and y 1 a 11 a 12 a 13 a 14 b 1 only if y 2 a 21 a 22 a 23 a 24 b 2 x j t j = 0 for all j = 1 , . . . , n y 3 a 31 a 32 a 33 a 34 b 3 c 1 c 2 c 3 c 4 y i s i = 0 for all i = 1 , . . . , m Consequences of Duality 30/40
Interpretation of Complementary Slackness Economic Interpretation Given an optimal primal production vector x and optimal dual offer prices y , Facility produces only products for which it is indifferent between sale and production. Only raw materials that are binding constraints on production are priced greater than 0 Consequences of Duality 31/40
Interpretation of Complementary Slackness Physical Interpretation Only walls adjacent to the balls equilibrium position push back on it. Consequences of Duality 31/40
Proof of Complementary Slackness Primal LP Dual LP maximize c ⊺ x minimize y ⊺ b subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Consequences of Duality 32/40
Proof of Complementary Slackness Primal LP Dual LP maximize c ⊺ x minimize y ⊺ b subject to Ax + s = b subject to A ⊺ y − t = c x � 0 y � 0 s � 0 t � 0 Can equivalently rewrite LP using slack variables Consequences of Duality 32/40
Proof of Complementary Slackness Primal LP Dual LP maximize c ⊺ x minimize y ⊺ b subject to Ax + s = b subject to A ⊺ y − t = c x � 0 y � 0 s � 0 t � 0 Can equivalently rewrite LP using slack variables y ⊺ b − c ⊺ x = y ⊺ ( Ax + s ) − ( y ⊺ A − t ⊺ ) x = y ⊺ s + t ⊺ x Consequences of Duality 32/40
Proof of Complementary Slackness Primal LP Dual LP maximize c ⊺ x minimize y ⊺ b subject to Ax + s = b subject to A ⊺ y − t = c x � 0 y � 0 s � 0 t � 0 Can equivalently rewrite LP using slack variables y ⊺ b − c ⊺ x = y ⊺ ( Ax + s ) − ( y ⊺ A − t ⊺ ) x = y ⊺ s + t ⊺ x Gap between primal and dual objectives is 0 if and only if complementary slackness holds. Consequences of Duality 32/40
Recovering Primal from Dual Will encounter LPs where the dual is easier to solve than primal Complementary slackness allows us to recover the primal optimal from the dual optimal, and vice versa. Consequences of Duality 33/40
Recovering Primal from Dual Will encounter LPs where the dual is easier to solve than primal Complementary slackness allows us to recover the primal optimal from the dual optimal, and vice versa. Assuming non-degeneracy: At every vertex of primal [dual] there are exactly n [ m ] tight constraints which are linearly independent. Consequences of Duality 33/40
Recovering Primal from Dual Will encounter LPs where the dual is easier to solve than primal Complementary slackness allows us to recover the primal optimal from the dual optimal, and vice versa. Assuming non-degeneracy: At every vertex of primal [dual] there are exactly n [ m ] tight constraints which are linearly independent. Primal LP Dual LP ( n variables, m + n constraints) ( m variables, m + n constraints) maximize c ⊺ x minimize y ⊺ b subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Consequences of Duality 33/40
Recovering Primal from Dual Will encounter LPs where the dual is easier to solve than primal Complementary slackness allows us to recover the primal optimal from the dual optimal, and vice versa. Assuming non-degeneracy: At every vertex of primal [dual] there are exactly n [ m ] tight constraints which are linearly independent. Primal LP Dual LP ( n variables, m + n constraints) ( m variables, m + n constraints) maximize c ⊺ x minimize y ⊺ b subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Let y be dual optimal. By non-degeneracy: Exactly m of the m + n dual constraints are tight at y Exactly n dual constraints are loose Consequences of Duality 33/40
Recovering Primal from Dual Will encounter LPs where the dual is easier to solve than primal Complementary slackness allows us to recover the primal optimal from the dual optimal, and vice versa. Assuming non-degeneracy: At every vertex of primal [dual] there are exactly n [ m ] tight constraints which are linearly independent. Primal LP Dual LP ( n variables, m + n constraints) ( m variables, m + n constraints) maximize c ⊺ x minimize y ⊺ b subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Let y be dual optimal. By non-degeneracy: Exactly m of the m + n dual constraints are tight at y Exactly n dual constraints are loose n loose dual constraints impose n tight primal constraints Consequences of Duality 33/40
Recovering Primal from Dual Will encounter LPs where the dual is easier to solve than primal Complementary slackness allows us to recover the primal optimal from the dual optimal, and vice versa. Assuming non-degeneracy: At every vertex of primal [dual] there are exactly n [ m ] tight constraints which are linearly independent. Primal LP Dual LP ( n variables, m + n constraints) ( m variables, m + n constraints) maximize c ⊺ x minimize y ⊺ b subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Let y be dual optimal. By non-degeneracy: Exactly m of the m + n dual constraints are tight at y Exactly n dual constraints are loose n loose dual constraints impose n tight primal constraints Assuming non-degeneracy, solving the linear equation yields a unique primal optimum solution x . Consequences of Duality 33/40
Sensitivity Analysis Primal LP Dual LP c ⊺ x y ⊺ b maximize minimize subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Sometimes, we want to examine how the optimal value of our LP changes with its parameters c and b Consequences of Duality 34/40
Sensitivity Analysis Primal LP Dual LP c ⊺ x y ⊺ b maximize minimize subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Sometimes, we want to examine how the optimal value of our LP changes with its parameters c and b Sensitivity Analysis Let OPT = OPT ( A, c, b ) be the optimal value of the above LP . Let x and y be the primal and dual optima. ∂OPT = x j when x is the unique primal optimum. ∂c j ∂OPT = y i when y is the unique dual optimum. ∂b i Consequences of Duality 34/40
Sensitivity Analysis Primal LP Dual LP c ⊺ x y ⊺ b maximize minimize subject to Ax � b subject to A ⊺ y � c x � 0 y � 0 Sometimes, we want to examine how the optimal value of our LP changes with its parameters c and b Economic Interpretation of Sensitivity Analysis A small increase δ in c j increases profit by δ · x j A small increase δ in b i increases profit by δ · y i y i measures the “marginal value” of resource i for production Consequences of Duality 34/40
Outline Linear Programming Basics 1 Duality and Its Interpretations 2 3 Properties of Duals Weak and Strong Duality 4 5 Formal Proof of Strong Duality of LP Consequences of Duality 6 More Examples of Duality 7
Shortest Path Given a directed network G = ( V, E ) where edge e has length ℓ e ∈ R + , find the minimum cost path from s to t . 2 5 0 3 1 1 1 2 s t 0 3 3 1 2 2 More Examples of Duality 35/40
Shortest Path 2 5 0 3 1 1 1 2 s t 0 3 3 1 2 2 Primal LP Dual LP � e ∈ E ℓ e x e min max y t − y s s.t. � x e − � x e = δ v , ∀ v ∈ V. s.t. y v − y u ≤ ℓ e , ∀ ( u, v ) ∈ E. e → v v → e x e ≥ 0 , ∀ e ∈ E. Where δ v = − 1 if v = s , 1 if v = t , and 0 otherwise. More Examples of Duality 35/40
Shortest Path 2 5 0 3 1 1 1 2 s t 0 3 3 1 2 2 Primal LP Dual LP � e ∈ E ℓ e x e min max y t − y s s.t. � x e − � x e = δ v , ∀ v ∈ V. s.t. y v − y u ≤ ℓ e , ∀ ( u, v ) ∈ E. e → v v → e x e ≥ 0 , ∀ e ∈ E. Where δ v = − 1 if v = s , 1 if v = t , and 0 otherwise. Interpretation of Dual Stretch s and t as far apart as possible, subject to edge lengths. More Examples of Duality 35/40
Maximum Weighted Bipartite Matching Set B of buyers, and set G of goods. Buyer i has value w ij for good j , and interested in at most one good. Find maximum value assignment of goods to buyers. More Examples of Duality 36/40
Maximum Weighted Bipartite Matching Primal LP Dual LP � max w ij x ij � u i + � min p j i,j i ∈ B j ∈ G � s.t. x ij ≤ 1 , ∀ i ∈ B. s.t. u i + p j ≥ w ij , ∀ i ∈ B, j ∈ G. j ∈ G u i ≥ 0 , ∀ i ∈ B. � x ij ≤ 1 , ∀ j ∈ G. p j ≥ 0 , ∀ j ∈ G. i ∈ B x ij ≥ 0 , ∀ i ∈ B, j ∈ G. More Examples of Duality 36/40
Maximum Weighted Bipartite Matching Primal LP Dual LP � max w ij x ij � u i + � min p j i,j i ∈ B j ∈ G � s.t. x ij ≤ 1 , ∀ i ∈ B. s.t. u i + p j ≥ w ij , ∀ i ∈ B, j ∈ G. j ∈ G u i ≥ 0 , ∀ i ∈ B. � x ij ≤ 1 , ∀ j ∈ G. p j ≥ 0 , ∀ j ∈ G. i ∈ B x ij ≥ 0 , ∀ i ∈ B, j ∈ G. Interpretation of Dual p j is price of good j u i is utility of buyer i Complementary Slackness: A buyer i only grabs goods j maximizing w ij − p j Only fully assigned goods have non-zero price A buyer witn nonzero utility must receive an item More Examples of Duality 36/40
2-Player Zero-Sum Games Rock-Paper-Scissors R P S R 0 1 − 1 P − 1 0 1 S 1 − 1 0 Two players, row and column Game described by matrix A When row player plays pure strategy i and column player plays pure strategy j , row player pays column player A ij More Examples of Duality 37/40
2-Player Zero-Sum Games Rock-Paper-Scissors R P S R 0 1 − 1 P − 1 0 1 S 1 − 1 0 Two players, row and column Game described by matrix A When row player plays pure strategy i and column player plays pure strategy j , row player pays column player A ij Mixed Strategy: distribution over pure strategies More Examples of Duality 37/40
2-Player Zero-Sum Games Rock-Paper-Scissors R P S R 0 1 − 1 P − 1 0 1 S 1 − 1 0 Two players, row and column Game described by matrix A When row player plays pure strategy i and column player plays pure strategy j , row player pays column player A ij Mixed Strategy: distribution over pure strategies If one of the players moves first, the other observes his mixed strategy but not the outcome of his coin flips. More Examples of Duality 37/40
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