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Can one employ convex optimization for symmetric but not positive definite problems? I tried using MATLAB's quadprog() function to solve this problem:

$\mathrm{min}\frac{1}{2}\mathbf{x}^T\mathbf{H}\mathbf{x}+\mathbf{x}^{T}\mathbf{g}$

$\mathrm{s.t.} \mathbf{Ax}=\mathbf{b}$

Where $\mathbf{H}$ is a symmetric but not positive definite. I am attempting to solve Darcy's equation based on the Variational Multi-Scale (VMS) formulation. I could rewrite the weak form such that the linear operator is symmetric, but even so quadprog will not work.

However, if I "normalized" the minimization functional:

$\mathrm{min}\frac{1}{2}\mathbf{x}^T\mathbf{H}^{T}\mathbf{H}\mathbf{x}+\mathbf{x}^{T}\mathbf{H}^T\mathbf{g}$

$\mathrm{s.t.} \mathbf{Ax}=\mathbf{b}$

I ensure that my Hessian is symmetric and positive definite, and quadprog does what i want it to. Why is this the case?

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A quadratic objective function in which the Hessian isn't positive semidefinite isn't a convex objective function and minimizing it isn't convex optimization.

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  • $\begingroup$ So if the Hessian is positive semi-definite the convex optimization will work, but if it's indefinite (e.g., a saddle-point problem) then it wont work? $\endgroup$ – Justin Aug 31 '15 at 23:37

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