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Results tagged with nonlinear-optimization
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user 32507
Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.
1
vote
Accepted
Are the intersection of proximinal sets in a Hilbert Space proximinal?
This answer is strongly inspired by example 3.11 in the book by Bauschke and Combettes.
Let $H = \ell^2$ and consider a sequence $\{\alpha_n\} \in (1,\infty)$ with $\alpha_n \searrow 1$. Define
\begi …
- 1,724
0
votes
Optimality condition for strongly convex function under sparsity constraint
I don't think that this is true. Let us take $p = 2$, $s = 1$ and $f(x) = \frac12 \|x - (1,1)\|^2$. Then, $\theta_0 = (1,0)$ is a minimizer, but with $\theta = (0,1)$ we get
$$
\nabla f(\theta_0)^\top …
- 1,724
1
vote
Hardness of concave minimization problem
If your problem has a solution $x^* \ne 0$, then $0$ is also a solution. Indeed, consider the function
$$\varphi(t) = c(t \, x^*) - k\cdot (t \, x^*).$$
Since $x^*$ is a solution, we have
$$\varphi(0) …
- 1,724
1
vote
Is a Lipschitz continuous gradient equivalent to this condition?
Yes, the converse is also true. This follows from the answer in https://math.stackexchange.com/questions/4227159/characterization-of-lipschitz-derivative.
In fact, your condition yields
$$
| (\nabla f …
- 1,724