9
votes
Accepted
Non-symmetric Positive Definite/Semidefinite Matrix in Quadratic Program
Yes, a real PSD matrix $M$ is a symmetric matrix with $$x^TMx\ge 0$$ for any $x$ (see e.g. https://en.wikipedia.org/wiki/Definite_matrix).
However, this is not a real restriction. (We have two ...
- 2,585
5
votes
Accepted
Augmented Lagrangian Function for Semidefinite Programming Problems
My way of reading it is $\langle X, \mathcal{A}^*(y)+S-C\rangle = \langle X, \mathcal{A}^*(y)-C\rangle + \langle X,S \rangle$. The first term is your standard inner product between dual variable and ...
- 1,702
4
votes
Augmented Lagrangian Function for Semidefinite Programming Problems
There's a good discussion of this in Convex Optimization by Stephen Boyd and Lieven Vandenberghe. See section 5.9.
With an ordinary scalar inequality constraint:
$f_{i}(x) \leq 0$,
you'll have a term ...
- 801
3
votes
Accepted
Conditions required for strong duality to hold for SDPs
The following is from Section 2.2 of Semidefinite Programming for Combinatorial Optimization by Christoph Helmberg.
Let's define
\begin{align*}
p^* &= \inf\{\langle C,X\rangle\,:\,X\text{ primal ...
1
vote
Solver for nonlinear semidefinite optimization
If you use Python, I have found calling the scs solver through cvxpy to be very effective for semidefinite programming. The <...
- 613
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