Questions tagged [non-smooth-optimization]

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Proof that generalized directional derivative is upper semicontinuous

In "Nonsmooth Optimization" by Mäkela and Neittaanmäki the definition of the generalized directional derivative is given as follows: Definition 3.1.1 (Clarke). Let $f: \mathbf{R}^{n} \...
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17 views

What is lower C1 regularity?

In the context of nonsmooth analysis and optimization, what is lower C1 regularity? I have found a definition of subsmoothness/ lower-C1 but cannot find anything about lower C1 regularity. Many thanks....
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1answer
34 views

Equivalence of the two expression for dual averaging

I have a question about two versions of the algorithm for dual averaging. The first one is $$x_{t+1}=x_t+\nabla f(w_t)\to w_{t+1}=P_C\left(-\frac{1}{\sqrt{t+1}}x_{t+1}\right)=\arg\min_{w\in C}\left\|w+...
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12 views

Reference request for composite optimization.

I need to cite a reference for composite optimization problem. My text goes like: "... The problem of minimizing the sum of a smooth function $f$ and a non-differentiable function $g$ over a ...
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0answers
24 views

Sub-differential of a convex function along a particular direction

Take a convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. Choose an arbitrary direction $d \in \mathbb{R}^n$ and consider the restriction of $f$ to the line through $x \in \mathbb{R}^n$ in the ...
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1answer
53 views

How to avoid overflow when evaluating the exponential smoothing function?

The exponential smoothing function is $f:\Bbb R^n\to \Bbb R$ defined as $$f(x):= \log\left(\sum_{i=1}^{n}e^{x_i}\right).$$ Obviously, when $x_i$ is large for some $i$, the term inside the logarithmic ...
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1answer
43 views

Minimizing a composite non-differentiable convex function over a $2$-norm ball

I am searching for (works on) methods for solving the composite differentiable and non-differentiable convex problem: $$ \min_{x \in B} f(x) + g(x),$$ where $B$ is a $2$-norm ball, ie: $x \in B \iff ...
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38 views

Formulate the solution of a zero sum game with a payoff matrix A as an optimization problem.

let $x\in$ ${\Bbb R}^n$ be the strategy of the first player and let $y\in$ ${\Bbb R}^n$ be the strategy of the second player. That is, $x_i$ represents the probability of the first player taking the ...
2
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1answer
200 views

Optimization with parametric constraints: solution maps

For constrained optimization problems $$ \begin{array}{ll} \min\limits_{x \in \mathbb R^n} & f(p, x) \\ \text{s.t.} & x \in C \end{array} $$ where $p \in \mathbb R$ is a parameter, we can ...
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78 views

Uses of nonsmooth analysis in mathematical research

To give some context: I am aware of the uses of Convex Analysis (and its applications in Convex Optimization), I have been studying (for a while) the developments of Nonsmooth Analysis (and its ...
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78 views

Second order necessary and sufficient conditions for convex nonsmooth optimization

For convex smooth optimization, first and second order necessary and sufficient conditions are well known. Does such standard second order necessary and sufficient conditions exist for convex ...
2
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1answer
66 views

Nonsmooth optimization approximation

Say I want to minimize a real valued, nonsmooth function $f(x)$ (gradient is not defined at some minima). Further let $$ g(x, \epsilon) $$ be a smooth approximation of $f(x)$ with $$ \lim_{\...
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2answers
492 views

Global optimization of non-smooth function

I have a number of functions (see for example two of them down below), and I need to find their global optimum for each of them. They are non-smooth, but they are always funnel-shaped, exhibiting a ...
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1answer
138 views

Real life nonsmooth convex optimization problem

I need to find a real world nonsmooth convex optimization problem, and determine the optimality conditions. What would be a basic problem that you would come across in the field, where I could ...
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79 views

convex optimization with multiple nonsmooth terms

Is there a general algorithm for solving $$ \min f(x) + g(x) + h(x) $$ where all three functions are convex and proximable, $f(x)$ is smooth, and $g(x)$ and $h(x)$ are both nonsmooth? Note that if ...
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660 views

KKT conditions for nonsmooth convex problems

What are the KKT conditions for a non-smooth convex function? Is the vanishing gradient of Lagrangian, replaced by $0$ in sub-differential of the Lagrangian, and all other things remain the same? I ...