This tag is for questions relating to subgradient, an iterative method for solving convex minimization problems, used predominantly in Nondifferentiable optimization for functions that are convex but nondifferentiable. The subgradient method is a very simple algorithm for minimizing convex nondifferentiable functions where newton's method and simple linear programming will not work.

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### Clarke regularity of piecewise linear functions

Let $f:X\rightarrow\mathbb{R}$, with $X\subseteq\mathbb{R}^n$, be a Lipschtiz continuous piecewise linear function. Is this sufficient to guarantee that $f$ is Clarke-regular (Definition 2.3.4 in [1])...
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1 vote
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### Example of convex function that does not have subgradient

Through self-study, I found out that there are some pathological functions that do not have subgradients. https://www.stat.cmu.edu/~siva/teaching/725/lec2.pdf On page 2: Except for some very ...
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### Swap integral and subgradient for convex functions

A very nice and helpful result is that for convex functions, the integral of a subgradient is always a subgradient of the integral, without any domination assumption needed. More precisely, let $T$ be ...
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### Same result in every iterations from subgradient and proximal gradient method.

I'm trying to implement the subgradient method and proximal gradient method with constant stepsize for the lasso problem but the result for the subgradient method and proximal gradient is almost ...
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• 724
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### Where does the $\ell_2$ norm subdifferential come from?

Looking at the definition of the subdifferential, we have that $v$ is a subdifferential of a function $f$ at a point $x$ if $$f(y) \geq f(x) + g^T(y-x), \forall y$$ Now, for $f(x) = \|x\|_2$, it's ...
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### If $p$ is bounded and bounded away from $0$, can we find upper and lower bound for $\|\nabla\hat p\|^2+\Delta\hat p$?

Let $d\in\mathbb R^d$ and $p:\mathbb R^d\to(0,\infty)$. Moreover, let $\sigma>0$, $$\tilde p(x):=p(\sigma x)\;\;\;\text{for }x\in\mathbb R^d$$ and $$\hat p:=\frac12\ln\tilde p.$$ Question: Can we ...
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### The steepest descent direction constrained to non-negative variables

Recently, another user asked about the steepest descent direction constrained to non-negative variables, but using the $L_1$ norm to avoid null directions, see this link. His ideas led to the ...
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### Fundamental Theorem of Calculus for non-differentiable convex functions

Suppose $C\subset\mathbb{R}^n$ is a convex set and $f:C\to\mathbb{R}$ is a convex function. I wonder if the following statement is true. Suppose $g:C\to\mathbb{R}^n$ satisfies $g(x)\in\partial f(x)$ ...
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### Relation Between Subgradients of a Random Function and Its Expectation

Suppose $\mathcal{X}\subset\mathbb{R}^n$ is a convex set. Let $f:\mathcal{X}\times\mathbb{R}^m\to\mathbb{R}$ be a function such that for every $y\in\mathbb{R}^m$, the function $f(\cdot,y)$ is convex, ...
• 709
1 vote
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### KKT conditions for non-differentiable constraints

So I know that for the problem: \begin{align*} \text{minimize} \quad & f_0(x) \\ \text{subject to} \quad & f_i(x) \leq 0, \quad i = 1, 2, \ldots, m \\ \end{align*} We have the following ...
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### Subdifferentials and optimality conditions of $l$-$0$ minimization

Consider the $l$-$0$ minimization problem: $$\min_{x} f(x) + \lambda \|x\|_0$$ First, prove that $g(x) = \|x\|_0$ is lower semicontinuous and then find the limiting subdifferential $\partial g(x)$ and ...
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