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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### Is a convex function with these properties possible?

Consider $V:[0,1]^2 \rightarrow[0,1]$, satisfying the following properties: $V(x,y) = p(x,y)x+(1-p(x,y))y$, where $p(x,y) \in [0,1]$. In other words, every point in $[0,1]^2$ is mapped to a ...
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### A subdifferential formulated as an argmax problem

I am reading the article "Random Variables, Monotone Relaitions and Convex Analysis" by Rockafellar and Royset. It is article number 226 on Rockafellar's website https://sites.math.washington.edu/~rtr/...
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### subgradient and subdifferential at the boundary of a closed set for discontinuous function. [closed]

Consider the function $f:[0,1] \to \mathbb{R}, \qquad$ $f(x) = \begin{cases} x^2 & \text{if$x > 0$,} \\ 1 & \text{if$x=0$.} \end{cases}$ This function is convex. I want to find ...
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### Proximal Mapping - Derivation of the Proximal Operator from the Resolvent of the Sub Differential

I do not really understand the solution for this question. I do not understand how the zero vector was derived to be an element of the subdifferential and why g(u) is strongly convex. Any help and ...
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### Need help solving a min-max fixed point equation

I have the following fixed point equation: for all $p\in[0,1]$ $$V(p) = \min_{\lambda_1,\lambda_2\in[0,1]}\max\{pr,(1-p)r,\beta\mathbb{E}_{a,y'}[V(f_{\lambda=(\lambda_1,\lambda_2)}(p,a,y'))]\}$$ ...
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I am trying to understand how can one compute subgradients for functions $f: \mathbb{R}^2 \supset M \to \mathbb{R}$. I know that $\xi \in \mathbb{R}^2$ is a subgradient for function $f$ in $x_0$ if ...
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### Computing subderivative for two variable function

I need to compute a suberivaqtive for the following function $$f\big((x, y) \big) = |x-2y+1| + |x-4y-3| + |2x-y+2|.$$ I also need to find its global extrema. I know that a subderivarive of function ...
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### Show that if $g(x) = f(Ax + b)$, then $\delta g(x) = A^T \delta f(Ax + b)$

Looking for a simplier expaination for the following: Show the following for sub-gradients: (a) If $g(x) = f(Ax + b)$, then $\delta g(x) = A^T \delta f(Ax + b)$. I've found the trivial ...
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### how to compute subgradient of p-norm, $1 \leq p \leq \infty$

I am a graduate student in Communications Engineering and I have an optimization course. I need help in solving my assignments: this one is related to convex optimization. We are asked to compute the ...
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### Subgradients of strictly Schur-convex function

Let $F: \mathbb{R}^N \to \mathbb{R}$ be strictly Schur-convex. It is well-known that if $F$ is differentiable, then for any $x=(x_1,\ldots,x_N)$ the following holds: for any $1\leq i,j\leq N$ such ...
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### An alternative to $\alpha_k=\frac{C}{k}$ step size

I am trying to implement a derivative of subgradient method that includes a distributed scenario and one of the requirements for convergence is to have a step size which satisifies the following ...
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### How do I take the inner product of a subdifferential with a vector?

I'm reading this pdf about an implementation of the PDHG algorithm for convex minimization. At the beginning of page 9, authors make an operation I can't not understand. In short, this is what ...
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### Proximal normal cone and convex sets

The proximal normal cone $N_S^P(x)$ for a set $S \subset X$, where $X$ is a Hilbert space, is defined as $$N_S^P(x) = \{\zeta \in X : d_S(x + t\zeta) = t\|\zeta\|, \text{ for some } t > 0\}.$$ ...
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### What will be the value of sub gradient at $0$ for function $|x|$

I am learning about Lasso Regression and came across taking gradient with respect to $0$. I came to know about subgradient but could not understand what will be its value at $0$. In lasso regression, ...
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### Gradient of $2$-norm inside absolute value

Compute the gradient of $$z = | \lVert L x \lVert_2 - P |$$ where $x \in \mathbb{C}^N$ and $P \in \mathbb{R}_{+}$. First, I make $a = \lVert L x \lVert_2 - P$. Also, since $| a |$ is non-...
consider a general convex function $f$ which is Lipschitz continuous over $X$, i.e., $\exists M > 0$ such that $$\left|f(x)-f(y)\right| \leq M\|x-y\|.$$ Here $X\subseteq R^n$is a closed ...