All Questions
68 questions
5
votes
1
answer
107
views
Optimization with weaker oracle than projection
I'm looking to solve the optimization problem
$$
min_{x \in C} ~ f(x),
$$
where $C \subset R^n$ is a closed, convex, bounded set and $f : R^n \to R$ a Lipschitz differentiable (nonconvex) function.
...
- 303
5
votes
1
answer
581
views
Strong duality for a particular moment problem
Reading the paper in this Link (see pag 13) with the objective of understanding a topic related to stochastic optimization I came across a problem in demonstrating one of the theorems. The situation ...
- 159
5
votes
1
answer
220
views
Analysis of first-order methods for constrained convex optimization with approximate oracles
In many first-order optimization methods an oracle is needed whose action enforces the constraint/regularizations. For example, in projected gradient descent, conditional gradient method, and proximal ...
- 215
4
votes
3
answers
200
views
Maximizing a pseudoconcave function in a box
I am trying to solve the problem:
$\max_{\boldsymbol{s}\in\mathbb{R}^{n}} \frac{\sqrt{\boldsymbol{a}^{T}\boldsymbol{s}+\alpha}}{\boldsymbol{b}^{T}\boldsymbol{s}+\beta}\\
\text{s.t} \;\;0\leq s_{i}\...
- 503
4
votes
1
answer
163
views
Gap to fill in the Aubin–Ekeland proof of the mountain-pass theorem
Working through the proof of the mountain-pass theorem given in Applied Nonlinear Analysis by Aubin & Ekeland, at what seems to be a critical point of the proof (the top of page 274) they refer to ...
- 193
4
votes
0
answers
241
views
Stochastic subgradient descent almost sure convergence
I was reading up on stochastic subgradient descent, and most sources i could find via google search give quick proofs on convergence in expectation and probability, and say that proofs of almost sure ...
- 141
3
votes
2
answers
266
views
Fixed point iteration on symmetric biconvex function
Suppose $X\subseteq\mathbb{R}^n$ is a convex set and that a function $g(x,y):X\times X\rightarrow\mathbb{R}_+$ is smooth, "strictly biconvex" (strictly convex in $x$ and $y$ independently but not ...
- 705
3
votes
1
answer
205
views
it's convex sequence inequality
A sequence $a_0,a_1,\dots,a_n$ of real numbers is called concave if $a_{0}=0$, and for each $0<i<n$, we have $a_i\geq\dfrac{a_{i-1}+a_{i+1}}{2}$.
Find the largest $c(n)$ such that for every ...
- 4,280
3
votes
1
answer
189
views
Sensitivity of the solution of QP with respect to parameters
Given a quadratic program,
$$\begin{array}{ll} \text{minimize} & \displaystyle \frac12 x^TAx + b^Tx \\ \text{subject to} & Cx \le d \end{array}$$
Suppose $A \succ 0$, so the program strongly ...
- 33
3
votes
2
answers
97
views
Optimal covering of line subsegments using a given set of disks
Is there a way of picking a minimal set of disks that's still covering the same line subsegments as all the disks together? Any help or references highly appreciated. Below is just an illustrative ...
3
votes
1
answer
803
views
Maximum of sum of exponential function
Let $x_1,\dots,x_n$ be a set of given vectors in $\mathbb{R}_{+}^d$. Let $c_1,\dots,c_n$ be given positive constants. I am interested in finding the vectors $w_1,\dots,w_n$ in $\mathbb{R}_{+}^d$ that ...
- 1,421
3
votes
1
answer
515
views
Optimizing input of an unknown function
Suppose we have a machine which takes the input $x_{in}$. In this machine the variable $x_{in}$ is converted to $y_{in}$ with the function $f(x)$, $f(x_{in})=y_{in}$. $f(x)$ is a known function, but ...
- 31
3
votes
1
answer
195
views
Partial results on composition of operators such that overall composition is monotone
(Adapted from Rockafellar)
Definition: Let $H$ be a real Hilbert space with inner product $\langle \cdot
,\cdot \rangle$. A function $T: H \to H$ is said to be a monotone
operator if \begin{...
- 155
3
votes
1
answer
2k
views
Global minimum of nonlinear least square
We have a continuous and differentiable function $f(\cdot)$ that maps from $R^n$ to $R^n$. We are trying to solve a nonlinear least square problem:
Minimize $J(x)=\Vert f(x)-z\Vert^2$
subject to box ...
- 33
3
votes
0
answers
91
views
What is the name for this type of optimization problem?
As we all know, a classic optimization problem can be represented in the following way:
Given a function $f: A \to \mathbb{R}$, find an element $x_0 \in A$ such that $f(x_0) \le f(x)$ for all $x \in ...
- 141
3
votes
0
answers
122
views
Convex optimization upper bound for a non-linear optimization
Is there any good convex optimization problem based upper-bound for the following non-linear optimization problem?
\begin{align}
\max_{x_1,\ldots,x_N}&\quad \sum_{n=1}^{N} \log(1+\frac{x_n}{1+\...
- 287
3
votes
0
answers
240
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 ...
- 791
3
votes
0
answers
255
views
How can we solve this kind of saddle point problem?
I'm trying to solve a saddle point problem of the following form: Let
$(E,\mathcal E,\lambda)$ be a measure space;
$p$ be a probability density on $(E,\mathcal E,\lambda)$ and $\mu:=p\lambda$
$W$ be ...
- 167
3
votes
0
answers
239
views
Constrained optimization with a Proportional-Integral-Derivative (PID) controller
My engineering colleagues have devised an interesting approach to equality-constrained optimization. I.e. they wish to solve the problem $\min_x f(x)$ subject to the constraint $g(x) = 0$ where $f, g ...
- 163
2
votes
0
answers
47
views
Why not use global optimization algorithms like PSO to solve decentralized control problems?
I do not see many works that use global optimization algorithms to solve decentralized control problems. Here the decentralized control problem means some entries of the feedback matrix are ...
- 21
2
votes
0
answers
171
views
How to sweep the leaves efficiently?
A cleaner, denoted by $P$, aims to sweep $n\ge 1$ leaves that appear one by one in a courtyard modeled by a compact set $D\subset \mathbb R^2$. Denote by $x_0$ the initial position of $P$ and by $v>...
user128095
2
votes
0
answers
46
views
Notion of distance between linear programs
Consider the linear programming problem
\begin{align}
\max_{x}&~c^Tx \\~s.t.~~a^Tx &\leq B~,~0\leq x_i \le1
\end{align}
where $c$ and $a$ are $n \times 1$ given non-negative vectors. $B$ is a ...
- 1,421
2
votes
0
answers
406
views
Pros and cons of using integer programming alone or combined integer and global optimization?
First, I am not sure if this is the right question to ask in this forum. But I have been looking for answers for a long time and I have been also asking my university's "engineering" professors but I ...
2
votes
0
answers
52
views
Zeroth order method with near-optimal rate that works in practice?
I want to find a ZO (zeroth-order, i.e. no access to gradient) algorithm to minimize a strongly-convex deterministic objective (say, as a sum of smooth and nonsmooth proximable functions). I want such ...
- 41
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 $$S:=\left\{w\in{\mathcal L^...
- 167
2
votes
0
answers
64
views
Finding orthogonal basis with constraint
Is there any fast algorithm that output an orthogonal basis $e_i,i\leq n$ of $R^n$
with $e_i\in V_i$? Where $V_i,i\leq n$ are given linear subspaces of $R^n$.
And is there any condition on $V_i,i\leq ...
- 909
2
votes
0
answers
242
views
Quadratic optimization with parameter in constraint
Disclaimer: I posted the same question on math.stackexchange. However, the FAQ suggests to post research-level questions in this forum.
Question: Given a function $q: \mathbb R^{N\times N}\mapsto \...
1
vote
1
answer
84
views
optimization over moving domains
Let $A, B$ be Banach spaces, and for any $a\in A$, $B_a\in B$ is a measurable subset. Consider the following optimization problem:
$$L(a)=\inf_{b\in B_a}\ell(b),$$
where $\ell(b)$ is a infinite-times ...
- 87
1
vote
1
answer
234
views
Constrained optimization of sum of squares polynomials
Consider the problem
$$
\min p(x) \text{ subject to } g_j(x)\le 0
\quad
p,g_j\in\text{SOS},
\qquad
(*)
$$
i.e. $p,g_j$ ($j=1,\ldots,m$) are sum of squares (SOS) polynomials. Can this problem be ...
- 710
1
vote
2
answers
270
views
Can we substitute this KKT condition into this optimization problem to reformulate the optimization problem?
Suppose I have the following optimization problem
$$ \min\limits_{\mathbf{x},\mathbf{y}} f(\mathbf{x},\mathbf{y}) \tag{1} $$
It is already known that the target function $f$ is continuous and ...
- 73
1
vote
1
answer
148
views
How can we calculate the generalized gradient of $L^2\ni x\mapsto a\min(x(s),by(t))$?
Let $(T,\mathcal T,\tau)$ be a measure space, $a,b\ge0$, $s,t\in T$ and $$f(x):=a\min(x(s),bx(t))\;\;\;\text{for }x\in L^2(\tau).$$
How can we calculate the generalized gradient $\partial_Cf(x)$ of ...
- 167
1
vote
1
answer
791
views
Hardness of concave minimization problem
I have an optimization problem $\underset{x}{\min} ~ c(x) - k \cdot x$ where $c(x)$ is a non-decreasing concave function with $c(0) = 0$, $x \in C \subseteq \mathbb{R}^d_{\geq 0}$. By non-decreasing, ...
- 29
1
vote
1
answer
336
views
A close-form solution for a simple quadratic optimization problem
Is there any closed form solution for the following optimization problem:
\begin{align}
&\min_{\mathbf{X},\alpha} \mathrm{Tr}[(\mathbf{A}-\mathbf{B}\mathbf{X})(\mathbf{A}-\mathbf{B}\mathbf{X})^{\...
- 287
1
vote
1
answer
189
views
Fritz-John conditions: Equality-constrained case as special case of inequality constraints
In Chapter 4 of Nonlinear Programming: Theory and Algorithms by Bazarra, Sherali, and Shetty, the following claim is made after Theorem 4.3.2 (Fritz-John necessary conditions):
"Note also that these ...
- 11
1
vote
1
answer
411
views
No strong duality In spite of Slater's condition
I was reading some course notes here.
On Page 8, it says:
Note that strong duality holds here (Slater's condition), but the
optimal value of the last problem is not necessarily the optimal
...
- 11
1
vote
0
answers
29
views
Change in active constraints when perturbing the objective of a QP
Suppose I have a quadratic program (with positive semidefinite cost matrix) with affine (polytopic) constraints. It is known that the solution to this is piecewise affine, with the ``pieces'' defined ...
- 11
1
vote
0
answers
71
views
LICQ vs MFCQ who is stronger [closed]
I want to ask you which constraint is stronger: MFCQ or LICQ.
- 11
1
vote
0
answers
97
views
How to solve the following optimization problem?
Let $G=(V,E)$ be a connected network with $|V|=n$. Consider the following optimization problem
I'm trying to know under which conditions the following minimization problem has solution :
$${\sum _{i=1}...
- 47
1
vote
0
answers
41
views
Fitting a non-periodic sum of periodic time series
The problems is as follows: you have $n$ points $(x_1,y_1),\dots,(x_n,y_n)$ and you want to fit the following equation to the data points:
$$y=\theta_1\cos(\theta_2 x+\theta_3) + \theta_4\cos(\theta_5 ...
- 3,098
1
vote
0
answers
98
views
Solution of a simple optimization problem
Let $\mathbf{U}_1$ and $\mathbf{U}_2$ be two arbitrary unitary matrices and $\mathbf{D}$ be a diagonal matrix. What is the solution of the following optimization problem?
\begin{align}
\min_{\mathbf{...
- 287
1
vote
2
answers
278
views
Optimization of a integral function
I have a function $h(y,x_1,x_2,\ldots,x_n)$. It is known that the minimum value of $h$ for any $y$ is attained when $x_1 = x_n$ and $x_2 = x_3 = \cdots = x_{n-1}$. Now consider the following function
\...
- 211
1
vote
0
answers
59
views
Minimizing square roots with the consecutive ones property
Let $A=[a_{ik}]$ be a matrix with the consecutive ones property in each column, i.e. each column consists of a single consecutive block of $1$'s (with zeros everywhere else). Is there anything at all ...
- 4,049
1
vote
0
answers
140
views
Factorization of argmax
We consider a function $f(s_{1:p}, a_{1:p})$, where $p>1$ is an integer, $s_{1:p}$ denotes $(s_1,\ldots,s_p)^\top \in R^p$, and $a_{1:p}$ denotes $(a_1,\ldots,a_p)^\top \in R^p$.
Question: What is ...
- 1,127
1
vote
1
answer
211
views
Does coercivity/supercoercivity conjugates?
According to Wikipedia, a function $f: \mathbb{R}^n \to \mathbb{R} \cup \{-\infty, +\infty\}$ is called coercive if,
$$f(x) \to +\infty \text{ as } \|x\| \to +\infty$$
and it is super-coercive if
$$\...
- 125
1
vote
0
answers
34
views
Are such assumptions of functions similar to strong convexity reasonable in convex optimization?
For $\mu$-strongly convex function $f:\mathbb{R}^d\to\mathbb{R}$, the following property holds: for any given $x,y\in\mathbb{R}^d$, we have
$$
(\nabla f(x) - \nabla f(y))^\top(x-y) \ge \mu \|x-y\|^2.$$...
- 97
1
vote
0
answers
48
views
Various definitions of coercivity
In this post one says that a functional $F:H\rightarrow [0,\infty]$ on an infinite-dimensional Hilbert space $H$ is (strongly) coercive if there exists a constant $k>0$ such that
$$
F(x)\geq k\|x\|...
- 5,405
1
vote
0
answers
79
views
Minimization of a smooth integral functional over a closed convex set
Let $(E,\mathcal E,\mu)$ be a probability space, $I$ be a finite nonempty set, $\gamma:(E\times I)^2\to[0,\infty)$ be measurable, $$F_1(g,w):=\sum_{i\in I}\int\mu({\rm d}x)w_i(x)g(x)\sum_{j\in I}\int\...
- 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
0
answers
188
views
Solution to a Strongly Convex Non-smooth Minimization Problem involving an L1 Norm
Let $X \in \mathbb{R}^{n \times d}, w \in \mathbb{R}^d, y \in \{\pm 1\}^{n}, \alpha \in [0,1], \lambda \in \mathbb{R}$. I have an expression that looks as follows
$\frac{1}{2}\|Xw -y \|_{2}^2 + \...
- 11
1
vote
0
answers
267
views
Minimum Preserving Transformations [closed]
If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then
$$
\operatorname{argmin}_{x \in X} f(x)
=
\operatorname{argmin}_{x \in X} g\circ f(x) .
$$
X and Y ...
- 5,405