All Questions
Tagged with nonlinear-optimization convex-analysis
26 questions
0
votes
1
answer
104
views
Optimality condition for strongly convex function under sparsity constraint
Let $f: \mathbb{R}^p \to \mathbb{R}$ be a $2s$-sparse strongly smooth, $2s$-sparse strongly convex and twice differentiable function. In other words, there exists positive constants $\alpha, L >0$ ...
- 399
0
votes
0
answers
129
views
Primal optimal attained implies dual optimal attained
Given some optimization problem
$$\min_{x \in S \subset \mathbb{R}^n} f_0(x) \quad \text{s.t.} \quad f_i(x) \leq 0, \quad 1\leq i\leq m$$
we can find the dual problem
$$\max_{\lambda\in\mathbb{R}^m} g(...
- 275
1
vote
1
answer
209
views
Does the value function of a quadratic program stay convex when adding constraints?
I am interested in the value function of a quadratic program of the form
$$
v(y)=\min_x \frac{1}{2} x^\top Q(y) x,
$$
subject to a linear equality constraint
$$
E(y)x=d(y),
$$
and a linear inequality ...
- 379
3
votes
1
answer
249
views
Interesting question about the Thomson problem for arbitrary number of electrons
This question is crossposted from here I believe this is a pretty hard question and so I decided to repost the question in the Math Overflow forum. If there is something wrong with doing this, I am ...
- 51
1
vote
1
answer
1k
views
Is a Lipschitz continuous gradient equivalent to this condition?
I know if a function $f: \mathbb{R}^n \to \mathbb{R}$ is $L$-smooth, i.e. its gradient $\nabla f$ is $L$-Lipschitz continuous, then it satisfies the following inequality for any $x, x_0 \in \mathbb{R}^...
- 163
1
vote
2
answers
270
views
Can we substitute this KKT condition into this optimization problem to reformulate the optimization problem?
Suppose I have the following optimization problem
$$ \min\limits_{\mathbf{x},\mathbf{y}} f(\mathbf{x},\mathbf{y}) \tag{1} $$
It is already known that the target function $f$ is continuous and ...
- 73
4
votes
1
answer
336
views
Which set of functions admits the existence of the minimizer?
Let $a,b \in \mathbb R$ and consider the functional $J$ on $X$:
$$J[u] = \int_0^1 \left( (u'(x))^2 -a)^2 + b \ln (1+ u^2(x))\right) dx$$
Providing reasons specify if the $\inf J$ over $X$ is attained ...
- 159
1
vote
1
answer
190
views
Proof of extended version of non-random "almost supermartingale"
In this question, a non-random version of "almost supermartingale" theorem is proved.
Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder ...
- 45
1
vote
1
answer
271
views
Can we invoke "almost supermartingale" Theorem for deterministic sequences?
Perhaps stupid question.
Question: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems?
Attempt ...
- 45
2
votes
1
answer
304
views
On the Lipschitz continuity of $x \mapsto \arg\min_{c \in C}d(x,c)$ w.r.t Hausdorff distance
Let $C$ be a (nonempty) compact subset of euclidean $\mathbb R^n$, and consider the set-valued map $p_C:\mathbb R^n \to 2^C$ defined by
$$
p_C(x) = \{c \in C \mid \|x-c\| = \mbox{dist}(x,C)\},
$$
...
- 6,853
2
votes
1
answer
388
views
Normal cones and KKT conditions
I'm trying to understand a statement from the book "Perturbation Analysis of Optimization Problems", by Bonnans and Shapiro. Let me start by providing some context. In page 148, the authors ...
- 159
2
votes
0
answers
111
views
Maximization of an integral functional over a closed convex set
I want to maximize $$F(w):=\sum_{1\le i,\:j\le2}\int\lambda^{\otimes2}({\rm d}(x,y))\left(w_i(x)f_j(x,y)\wedge w_j(y)f_i(y,x)\right)g_{ij}(x,y)$$ over the closed convex set $$S:=\left\{w\in{\mathcal L^...
- 167
1
vote
0
answers
167
views
Gradient formula for Clarke's generalized gradient on a general Banach space
In Theorem 10.27 of the book Functional Analysis, Calculus of Variations and Optimal Control, there is the following gradient formula:
($\operatorname{co}$ deotes the convex hull).
Is there an ...
- 167
1
vote
1
answer
148
views
How can we calculate the generalized gradient of $L^2\ni x\mapsto a\min(x(s),by(t))$?
Let $(T,\mathcal T,\tau)$ be a measure space, $a,b\ge0$, $s,t\in T$ and $$f(x):=a\min(x(s),bx(t))\;\;\;\text{for }x\in L^2(\tau).$$
How can we calculate the generalized gradient $\partial_Cf(x)$ of ...
- 167
0
votes
1
answer
98
views
1D functional equation: solve for function with given expected value w.r.t normal density
Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation
$$
\begin{split}
\mathbb ...
- 6,853
1
vote
1
answer
75
views
Calculate $k:=\sup\left\{\left\|\theta\right\|_{*} \: |\: \ell^{*}(\theta)<\infty\right\}$ for a special $\ell$ function
We consider the function $\ell:\mathbb{R}^{m}\rightarrow \mathbb{R}$ given by
$$\ell(\xi):=-\max\left\{-\left\langle x,\xi\right\rangle+10 \tau, -51\left\langle x,\xi\right\rangle -40\tau \right\}$$
...
- 159
1
vote
0
answers
22
views
Find conditions over $U$ such that optimization over a convex cone generated by $U$ is equal to optimization over $U$ [duplicate]
Reading several pappers to prepare my thesis I found the following problem:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \max\limits_{x\in\mathcal{C}} & f(x) \\[...
- 159
8
votes
0
answers
210
views
Concavity of product and ratio of sums
Apologies if this question is not appropriate for MathOverflow. I have asked at Math.StackExchange without success.
Consider the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ defined as
$$
f(x)=\...
- 379
1
vote
0
answers
49
views
Minimization of convex functions on dense subspaces
I want to consider the Moreau envelope $\psi_j$ of a proper, convex, lower semicontinuous function $\psi$ over a Banach space $V$ on dense (finite dimensional) monotonically increasing subspaces $V_m$,...
- 187
1
vote
0
answers
355
views
Solution to a system of nonlinear equations using convex conjugate of log sum exp
I need to prove the following result:
There exists a unique solution to the system of equations
$$\alpha = \frac{I^T(\gamma e^{I\beta})}{\sum_{i=1}^{n}\gamma_ie^{(I\beta)_i}}$$
if and only if $\...
- 11
4
votes
3
answers
2k
views
Zero lambda, zero constraint in the complementary slackness condition of the Kuhn-Tucker problem
Complementary slackness condition in the KKT theorem states that:
$\lambda_i^*\geq0; \lambda_i^*h_i(x^*)=0 $
The usual reasoning goes like this: either constraint is clack $h_i(x^*)>0$ and then ...
- 71
1
vote
0
answers
232
views
Semi-convex problem and almost convex problem
I have a target function, I've computed its Hessian to check convexity, it has a positive-definite sub-matrix and small negative-definite sub-matrix and a kernel. Sometimes it is even better -- the ...
- 355
0
votes
1
answer
141
views
How to compute the direction of slowest ascent from the minimum of a strongly convex function?
Consider a twice differentiable strongly convex function $f:\mathbb{R}^n \rightarrow \mathbb{R^+}$ that attains its minimum value at the point $x^*$. I am wondering if one can compute a direction of ...
- 379
2
votes
5
answers
3k
views
Distance between two sets
Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem.
$$ \min \{||x-y|...
- 57
14
votes
1
answer
2k
views
An intuition for three different types of subgradients (proximal, regular, limiting)
I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking.
These subgradients are (assume $x \in$ ...
- 249
0
votes
1
answer
272
views
A certain type of quadratic problem.
I am interested in solving the following equality constrained quadratic (?) problem.
\begin{align}
\min_{u^{H}u=1}~(u^{H}A_1u) \\\
s.t.~ u^{H}A_2u=0
\end{align}
$A_1$ and $A_2$ are $N\times N$ ...
- 1,421