CS675: Convex and Combinatorial Optimization Fall 2014 Optimality Conditions for Convex Optimization Instructor: Shaddin Dughmi
Outline Optimality Conditions 1
Recall: Lagrangian Duality Primal Problem Dual Problem min f 0 ( x ) max g ( λ, ν ) s.t. s.t. f i ( x ) ≤ 0 , ∀ i = 1 , . . . , m. λ � 0 h i ( x ) = 0 , ∀ i = 1 , . . . , k. Optimality Conditions 0/6
Recall: Lagrangian Duality Primal Problem Dual Problem min f 0 ( x ) max g ( λ, ν ) s.t. s.t. f i ( x ) ≤ 0 , ∀ i = 1 , . . . , m. λ � 0 h i ( x ) = 0 , ∀ i = 1 , . . . , k. Weak Duality OPT ( dual ) ≤ OPT ( primal ) . Optimality Conditions 0/6
Recall: Lagrangian Duality Primal Problem Dual Problem min f 0 ( x ) max g ( λ, ν ) s.t. s.t. f i ( x ) ≤ 0 , ∀ i = 1 , . . . , m. λ � 0 h i ( x ) = 0 , ∀ i = 1 , . . . , k. Strong Duality OPT ( dual ) = OPT ( primal ) . Optimality Conditions 0/6
Dual Solution as a Certificate Primal Problem Dual Problem min f 0 ( x ) max g ( λ, ν ) s.t. s.t. f i ( x ) ≤ 0 , ∀ i = 1 , . . . , m. λ � 0 h i ( x ) = 0 , ∀ i = 1 , . . . , k. Dual solutions serves as a certificate of optimality If f 0 ( x ) = g ( λ, ν ) , and both are feasible, then both are optimal. Optimality Conditions 1/6
Dual Solution as a Certificate Primal Problem Dual Problem min f 0 ( x ) max g ( λ, ν ) s.t. s.t. f i ( x ) ≤ 0 , ∀ i = 1 , . . . , m. λ � 0 h i ( x ) = 0 , ∀ i = 1 , . . . , k. Dual solutions serves as a certificate of optimality If f 0 ( x ) = g ( λ, ν ) , and both are feasible, then both are optimal. If f 0 ( x ) − g ( λ, ν ) ≤ ǫ , then both are within ǫ of optimality. OPT(primal) and OPT(dual) lie in the interval [ g ( λ, ν ) , f 0 ( x )] Optimality Conditions 1/6
Dual Solution as a Certificate Primal Problem Dual Problem min f 0 ( x ) max g ( λ, ν ) s.t. s.t. f i ( x ) ≤ 0 , ∀ i = 1 , . . . , m. λ � 0 h i ( x ) = 0 , ∀ i = 1 , . . . , k. Dual solutions serves as a certificate of optimality If f 0 ( x ) = g ( λ, ν ) , and both are feasible, then both are optimal. If f 0 ( x ) − g ( λ, ν ) ≤ ǫ , then both are within ǫ of optimality. OPT(primal) and OPT(dual) lie in the interval [ g ( λ, ν ) , f 0 ( x )] Primal-dual algorithms use dual certificates to recognize optimality, or bound sub-optimality. Optimality Conditions 1/6
Complementary Slackness Primal Problem Dual Problem min f 0 ( x ) max g ( λ, ν ) s.t. s.t. f i ( x ) ≤ 0 , ∀ i = 1 , . . . , m. λ � 0 h i ( x ) = 0 , ∀ i = 1 , . . . , k. Facts If strong duality holds, and x ∗ and ( λ ∗ , ν ∗ ) are optimal, then x ∗ minimizes L ( x, λ ∗ , ν ∗ ) over all x . λ ∗ i f i ( x ∗ ) = 0 for all i . (Complementary Slackness) Optimality Conditions 2/6
Complementary Slackness Primal Problem Dual Problem min f 0 ( x ) max g ( λ, ν ) s.t. s.t. f i ( x ) ≤ 0 , ∀ i = 1 , . . . , m. λ � 0 h i ( x ) = 0 , ∀ i = 1 , . . . , k. Facts If strong duality holds, and x ∗ and ( λ ∗ , ν ∗ ) are optimal, then x ∗ minimizes L ( x, λ ∗ , ν ∗ ) over all x . λ ∗ i f i ( x ∗ ) = 0 for all i . (Complementary Slackness) Proof f 0 ( x ∗ ) = g ( λ ∗ , ν ∗ ) m k � � ≤ f 0 ( x ∗ ) + λ ∗ i f i ( x ∗ ) + ν ∗ i h i ( x ∗ ) i =1 i =1 ≤ f 0 ( x ∗ ) Optimality Conditions 2/6
Complementary Slackness Primal Problem Dual Problem min f 0 ( x ) max g ( λ, ν ) s.t. s.t. f i ( x ) ≤ 0 , ∀ i = 1 , . . . , m. λ � 0 h i ( x ) = 0 , ∀ i = 1 , . . . , k. Facts If strong duality holds, and x ∗ and ( λ ∗ , ν ∗ ) are optimal, then x ∗ minimizes L ( x, λ ∗ , ν ∗ ) over all x . λ ∗ i f i ( x ∗ ) = 0 for all i . (Complementary Slackness) Interpretation Lagrange multipliers ( λ ∗ , ν ∗ ) “simulate” the primal feasibility constraints Interpreting λ i as the “value” of the i ’th constraint, at optimality only the binding constraints are “valuable” Recall economic interpretation of LP Optimality Conditions 2/6
min f 0 ( x ) max g ( λ, ν ) s.t. s.t. f i ( x ) ≤ 0 , ∀ i = 1 , . . . , m. λ � 0 h i ( x ) = 0 , ∀ i = 1 , . . . , k. KKT Conditions When strong duality holds, the primal problem is convex, and the constraint functions are differentiable, x ∗ and ( λ ∗ , ν ∗ ) are optimal iff: x ∗ and ( λ ∗ , ν ∗ ) are feasible λ ∗ i f i ( x ∗ ) = 0 (Complementary Slackness) ▽ x L ( x ∗ , λ ∗ , ν ∗ ) = ▽ f 0 ( x ∗ )+ � m i ▽ f i ( x ∗ )+ � k i ▽ h i ( x ∗ ) = 0 i =1 λ ∗ i =1 ν ∗ Optimality Conditions 3/6
min f 0 ( x ) max g ( λ, ν ) s.t. s.t. f i ( x ) ≤ 0 , ∀ i = 1 , . . . , m. λ � 0 h i ( x ) = 0 , ∀ i = 1 , . . . , k. KKT Conditions When strong duality holds, the primal problem is convex, and the constraint functions are differentiable, x ∗ and ( λ ∗ , ν ∗ ) are optimal iff: x ∗ and ( λ ∗ , ν ∗ ) are feasible λ ∗ i f i ( x ∗ ) = 0 (Complementary Slackness) ▽ x L ( x ∗ , λ ∗ , ν ∗ ) = ▽ f 0 ( x ∗ )+ � m i ▽ f i ( x ∗ )+ � k i ▽ h i ( x ∗ ) = 0 i =1 λ ∗ i =1 ν ∗ Why are KKT Conditions Useful? Derive an analytical solution to some convex optimization problems Gain structural insights Optimality Conditions 3/6
Example: Equality-constrained Quadratic Program 1 minimize 2 x ⊺ Px + q ⊺ x + r subject to Ax = b KKT Conditions: Ax ∗ = b and Px ∗ + q + A ⊺ ν ∗ = 0 Simply a solution of a linear system with variables x ∗ and ν ∗ . Optimality Conditions 4/6
Example: Market Equilibria (Fisher’s Model) Buyers B , and goods G . Buyer i has utility u ij for each unit of good G . Buyer i has budget m i , and there’s one divisible unit of each good. Optimality Conditions 5/6
Example: Market Equilibria (Fisher’s Model) Buyers B , and goods G . Buyer i has utility u ij for each unit of good G . Buyer i has budget m i , and there’s one divisible unit of each good. Does there exist a market equilibrium? Prices p j on items, such that each player can buy his favorite bundle and the market clears. Optimality Conditions 5/6
Example: Market Equilibria (Fisher’s Model) Buyers B , and goods G . Buyer i has utility u ij for each unit of good G . Buyer i has budget m i , and there’s one divisible unit of each good. Does there exist a market equilibrium? Prices p j on items, such that each player can buy his favorite bundle and the market clears. Eisenberg-Gale Convex Program maximize � i m i log � j u ij x ij subject to � i x ij ≤ 1 , for j ∈ G. x � 0 Optimality Conditions 5/6
Example: Market Equilibria (Fisher’s Model) Buyers B , and goods G . Buyer i has utility u ij for each unit of good G . Buyer i has budget m i , and there’s one divisible unit of each good. Does there exist a market equilibrium? Prices p j on items, such that each player can buy his favorite bundle and the market clears. Eisenberg-Gale Convex Program maximize � i m i log � j u ij x ij subject to � i x ij ≤ 1 , for j ∈ G. x � 0 Using KKT conditions, we can prove that the dual variables corresponding to the item supply constraints are market-clearing prices! Optimality Conditions 5/6
Recommend
More recommend
