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4 votes

Is $X \mapsto \operatorname{tr}( DXAX^TD^T - DXCD - DC^TX^TD)$ a convex function?

Note that $f(X)=\operatorname{tr}(DXAX^TD)$ is mid-point convex because \begin{align} &\frac{f(X)+f(Y)}{2}-f\left(\frac{X+Y}{2}\right)\\ &=\operatorname{tr}\left(\dfrac{DXAX^TD+DYAY^TD}{2} -D\...
user1551's user avatar
  • 141k
1 vote
Accepted

How to bound scalarized gradient difference norm in terms of smoothness in convex optimization?

This is not possible. In fact, you can always insert $u = v$ and have $$ \| \alpha \nabla g(u) - \beta \nabla g(v) \| = |\alpha - \beta| \|\nabla g(u) \| $$ and it is not possible to bound this term ...
gerw's user avatar
  • 31.6k
1 vote

Conceptual difference between regularized and constrained optimization.

I think that conceptually the key difference is very direct - the first formulation does not impose an explicit constraint on the maximum allowable fitting error, but the second does. Regarding the ...
Alex Shtoff's user avatar
1 vote
Accepted

Computing a "Generalized" Sinkhorn distance between two discrete probability distributions: A bi-convex optimization model

Here I provide some observations that can help you to study your problem. The problem in the OP is the minimization of a bi-convex function over a convex set. Bi-convex optimization problems form a ...
Amir's user avatar
  • 8,176
1 vote

Resolvent of maximal-monotone operator

Existence follows from Rockafellar's surjectivity theorem: Suppose that $X$ is reflexive and $A \colon X \to 2^{X^*}$ is a maximal monotone operator. Then $$R(A+λ \mathcal F ) =X^*, \quad \forall \...
Evangelopoulos Foivos's user avatar

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