On the (Non-)Differentiability of the Optimal Value Function When the Optimal Solution is Unique

2013 ◽  
Author(s):  
Daisuke Oyama ◽  
Tomoyuki Takenawa
Keyword(s):  
Value Function ◽  
Optimal Solution ◽  
Optimal Value Function ◽  
Optimal Value

Hadamard Directional Differentiability of the Optimal Value Function of a Quadratic Programming Problem

2018 ◽  
Vol 35 (03) ◽  
pp. 1850012 ◽  
Author(s):  
Sainan Zhang ◽  
Liwei Zhang ◽  
Hongwei Zhang ◽  
Qingsong Duan
Keyword(s):  
Quadratic Programming ◽  
Value Function ◽  
Convex Sets ◽  
Compact Convex ◽  
Optimal Solution ◽  
Optimal Value Function ◽  
Directional Differentiability ◽  
Optimal Value ◽  
Hadamard Directional Differentiability ◽  
Wolfe Dual

In this paper, we consider the stability analysis of a convex quadratic programming (QP) problem and its restricted Wolfe dual when all parameters in the problem are perturbed. Based on the continuity of the feasible set mapping, we establish the upper semi-continuity of the optimal solution mappings of the convex QP problem and the restricted Wolfe dual problem. Furthermore, by characterizing the optimal value function as a min–max optimization problem over two compact convex sets, we demonstrate the Lipschitz continuity and the Hadamard directional differentiability of the optimal value function.

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