Was exploring quadratic programming optimization and for two types of problems the answers seemed to always equal. Is there an intuitive proof?
Problem 1:
- Minimize $\tfrac{1}{2} \mathbf{x}^T Q\mathbf{x}$
- Subject to $ A \mathbf{x} = \mathbf{b} $ and $\mathbf{x}>0$
- $ Q = n \mathbf{I}_{(n,n)} - \mathbf{1}_{(n,n)}$
- Call the solution $\widehat{\mathbf{x}}$
Problem 2:
- Define $ S_R = R \mathbf{I}_{(n,n)} - \mathbf{1}_{(n,n)}$ for scalar $0<R<n$
- Maximum R such that there exist a solution $\mathbf{x}$ such that $\tfrac{1}{2} \mathbf{x}^T S_R\mathbf{x}=0$
- Subject to $ A \mathbf{x} = \mathbf{b} $ and $\mathbf{x}>0$ (same as Problem 1)
- Call the solution of $\mathbf{x}$ for the maximum R to be $\widetilde{\mathbf{x}}$
Is it always true that $\widehat{\mathbf{x}}$ = $\widetilde{\mathbf{x}}$