Duality (II) Lijun Zhang zlj@nju.edu.cn http://cs.nju.edu.cn/zlj
Outline Saddle-point Interpretation Max-min Characterization of Weak and Strong Duality Saddle-point Interpretation Game Interpretation Optimality Conditions Certificate of Suboptimality and Stopping Criteria Complementary Slackness KKT Optimality Conditions Solving the Primal Problem via the Dual Examples Generalized Inequalities
Outline Saddle-point Interpretation Max-min Characterization of Weak and Strong Duality Saddle-point Interpretation Game Interpretation Optimality Conditions Certificate of Suboptimality and Stopping Criteria Complementary Slackness KKT Optimality Conditions Solving the Primal Problem via the Dual Examples Generalized Inequalities
More Symmetric Form Assume no equality constraint � sup 𝑀 𝑦, 𝜇 � sup 𝑔 � 𝑦 � � 𝜇 � 𝑔 � 𝑦 �≽� �≽� ��� � �𝑔 � 𝑦 𝑔 � 𝑦 � 0, 𝑗 � 1, … , 𝑛 ∞ otherwise Suppose � for some . Then, by � and �≽� � If � , then the optimal choice of is and �≽� �
More Symmetric Form Optimal Value of Primal Problem 𝑞 ⋆ � inf � sup 𝑀 𝑦, 𝜇 �≽� Optimal Value of Dual Problem 𝑒 ⋆ � sup inf � 𝑀 𝑦, 𝜇 �≽� Weak Duality sup inf � 𝑀 𝑦, 𝜇 � inf � sup 𝑀 𝑦, 𝜇 �≽� �≽� Strong Duality sup inf � 𝑀 𝑦, 𝜇 � inf � sup 𝑀 𝑦, 𝜇 �≽� �≽� Min and Max can be switched
A More General Form Max-min Inequality sup �∈� 𝑔 𝑥, 𝑨 � inf inf �∈� sup 𝑔 𝑥, 𝑨 �∈� �∈� � � For any and any � � Strong Max-min Property sup �∈� 𝑔 𝑥, 𝑨 � inf inf �∈� sup 𝑔 𝑥, 𝑨 �∈� �∈� Hold only in special cases
Outline Saddle-point Interpretation Max-min Characterization of Weak and Strong Duality Saddle-point Interpretation Game Interpretation Optimality Conditions Certificate of Suboptimality and Stopping Criteria Complementary Slackness KKT Optimality Conditions Solving the Primal Problem via the Dual Examples Generalized Inequalities
Saddle-point Interpretation is a saddle point for 𝑔 𝑥 �, 𝑨 � 𝑔 𝑥 �, 𝑨̃ � 𝑔 𝑥, 𝑨̃ , ∀𝑥 ∈ 𝑋, 𝑨 ∈ 𝑎 minimizes , maximizes 𝑔 𝑥 �, 𝑨̃ � inf �∈� 𝑔�𝑥, 𝑨̃� , 𝑔 𝑥 �, 𝑨̃ � sup 𝑔 𝑥 �, 𝑨 �∈� https: / / en.wikipedia.org/ wiki/ Saddle_point
Saddle-point Interpretation is a saddle point for 𝑔 𝑥 �, 𝑨 � 𝑔 𝑥 �, 𝑨̃ � 𝑔 𝑥, 𝑨̃ , ∀𝑥 � ∈ 𝑋, 𝑨̃ ∈ 𝑎 minimizes , maximizes 𝑔 𝑥 �, 𝑨̃ � inf �∈� 𝑔�𝑥, 𝑨̃� , 𝑔 𝑥 �, 𝑨̃ � sup 𝑔 𝑥 �, 𝑨 �∈� Imply the strong max-min property sup �∈� 𝑔 𝑥, 𝑨 � inf inf �∈� 𝑔�𝑥, 𝑨̃� � 𝑔 𝑥 �, 𝑨̃ �∈� 𝑔 𝑥, 𝑨 � 𝑔 𝑥 �, 𝑨̃ � sup 𝑔 𝑥 �, 𝑨 � inf �∈� sup �∈� �∈� ⇒ sup �∈� 𝑔 𝑥, 𝑨 � inf inf �∈� sup 𝑔 𝑥, 𝑨 �∈� �∈� ⇒ sup �∈� 𝑔 𝑥, 𝑨 � inf inf �∈� sup 𝑔 𝑥, 𝑨 �∈� �∈�
Saddle-point Interpretation is a saddle point for 𝑔 𝑥 �, 𝑨 � 𝑔 𝑥 �, 𝑨̃ � 𝑔 𝑥, 𝑨̃ , ∀𝑥 � ∈ 𝑋, 𝑨̃ ∈ 𝑎 minimizes , maximizes 𝑔 𝑥 �, 𝑨̃ � inf �∈� 𝑔�𝑥, 𝑨̃� , 𝑔 𝑥 �, 𝑨̃ � sup 𝑔 𝑥 �, 𝑨 �∈� ⋆ are primal and dual optimal ⋆ If ⋆ ⋆ points and strong duality holds, form a saddle-point. If is saddle-point, then is primal optimal, is dual optimal, and the duality gap is zero.
Outline Saddle-point Interpretation Max-min Characterization of Weak and Strong Duality Saddle-point Interpretation Game Interpretation Optimality Conditions Certificate of Suboptimality and Stopping Criteria Complementary Slackness KKT Optimality Conditions Solving the Primal Problem via the Dual Examples Generalized Inequalities
Continuous Zero-sum Game Two players The 1st player chooses , and the 2nd player selects Player 1 pays an amount to player 2 Goals Player 1 wants to minimize Player 2 wants to maximize Continuous game The choices are vectors, and not discrete
Continuous Zero-sum Game Player 1 makes his choice first Player 2 wants to maximize payoff and the resulting payoff is �∈� Player 1 knows that player 2 will follow this strategy, and so will choose to make as small as possible �∈� Thus, player 1 chooses argmin sup 𝑔 𝑥, 𝑨 �∈� �∈� The payoff �∈� sup inf 𝑔 𝑥, 𝑨 �∈�
Continuous Zero-sum Game Player 2 makes his choice first Player 1 wants to minimize payoff and the resulting payoff is �∈� Player 2 knows that player 1 will follow this strategy, and so will choose to make as large as possible �∈� Thus, player 2 chooses argmax �∈� 𝑔 𝑥, 𝑨 inf �∈� The payoff sup �∈� 𝑔 𝑥, 𝑨 inf �∈�
Continuous Zero-sum Game Max-min Inequality sup �∈� 𝑔 𝑥, 𝑨 � inf inf �∈� sup 𝑔 𝑥, 𝑨 �∈� �∈� Player 2 plays first Player 1 plays first Player 1 wants to minimize Player 2 wants to maximize It is better for a player to go second
Continuous Zero-sum Game Strong Max-min Property sup �∈� 𝑔 𝑥, 𝑨 � inf inf �∈� sup 𝑔 𝑥, 𝑨 �∈� �∈� Player 2 plays first Player 1 plays first Player 1 wants to minimize Player 2 wants to maximize There is no advantage to playing second
Continuous Zero-sum Game Strong Max-min Property sup �∈� 𝑔 𝑥, 𝑨 � inf inf �∈� sup 𝑔 𝑥, 𝑨 �∈� �∈� Player 2 plays first Player 1 plays first Saddle-point Property If is a saddle-point for (and ), then it is called a solution of the game 𝑥 � : the optimal strategy for player 1 𝑨̃ : the optimal strategy for player 2 No advantage to playing second
A Special Case � � Payoff is the Lagrangian; � Player 1 chooses the primal variable while player 2 chooses the dual variable The optimal choice for player 2, if she must choose first, is any dual optimal ⋆ The resulting payoff: 𝑒 ⋆ Conversely, if player 1 chooses first, his ⋆ optimal choice is any primal optimal The resulting payoff: 𝑞 ⋆ Duality gap: advantage of going second
Outline Saddle-point Interpretation Max-min Characterization of Weak and Strong Duality Saddle-point Interpretation Game Interpretation Optimality Conditions Certificate of Suboptimality and Stopping Criteria Complementary Slackness KKT Optimality Conditions Solving the Primal Problem via the Dual Examples Generalized Inequalities
Certificate of Suboptimality Dual Feasible A lower bound on the optimal value of the primal problem ⋆ Provides a proof or certificate Bound how suboptimal a given feasible ⋆ point is, without knowing the value of ⋆ � � 𝑦 is 𝜗 -suboptimal for primal problem ( 𝜇, 𝜉� is 𝜗 -suboptimal for dual
Certificate of Suboptimality Gap between Primal & Dual Objectives � Referred to as duality gap associated with primal feasible and dual feasible localizes the optimal value of the primal (and dual) problems to an interval ⋆ ⋆ � � The width of the interval is the duality gap If duality gap of is , then is primal optimal and is dual optimal
Stopping Criteria Optimization algorithms produce a � and dual sequence of primal feasible � � feasible for Required absolute accuracy: ��� A Nonheuristic Stopping Criterion � � � � ��� � Guarantees when algorithm terminates, is ��� -suboptimal
Stopping Criteria A Relative Accuracy ��� Nonheuristic Stopping Criteria If � 𝜇 � , 𝜉 � � 𝑦 � 𝑔 𝜇 � , 𝜉 � � 0, � 𝜗 ��� 𝜇 � , 𝜉 � or � 𝜇 � , 𝜉 � � 𝑦 � 𝑔 � 𝑦 � 𝑔 � 0, � 𝜗 ��� � 𝑦 � �𝑔 ⋆ Then , and the relative error satisfies � 𝑦 � � 𝑞 ⋆ 𝑔 � 𝜗 ��� |𝑞 ⋆ |
Outline Saddle-point Interpretation Max-min Characterization of Weak and Strong Duality Saddle-point Interpretation Game Interpretation Optimality Conditions Certificate of Suboptimality and Stopping Criteria Complementary Slackness KKT Optimality Conditions Solving the Primal Problem via the Dual Examples Generalized Inequalities
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