All Questions
Tagged with convex-optimization oc.optimization-and-control
215 questions
1
vote
1
answer
82
views
Solution to a quadratically constrained quadratic program with unit ball constraint
I am working on a quadratically constrained quadratic program (QCQP) of the form:$$ \min_{x} \quad \frac{1}{2} x^T P x + q^T x + r$$
$$ \text{subject to} \qquad x^{T}x \leq 1 $$
where $P \in S^{++}_{...
- 13
0
votes
0
answers
35
views
Describing the boundary of the feasible direction cone to a convex open subset of $\mathbb{R}^n$ at a boundary point: connection via subdifferential?
Let $U\subset \mathbb{R}^n$ be a convex, open set with nonempty boundary. Let $x_0\in \partial{U}.$ We can describe $U$ locally near $x_0$ as a super level set of a suitable continuous concave ...
- 1,467
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
0
votes
1
answer
57
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Self-concordant barrier for the epigraph of $f(x,y) = x^p y^{1-p}$?
The problem
Assume $p > 1$. Consider the function
$$f(x,y) = x^p y^{1-p}, \qquad x,y > 0.$$
Note that
$$
f'' = p(p-1)x^{p-2}y^{-1-p}
\begin{bmatrix}
y \\ & x
\end{bmatrix}
\begin{bmatrix}
1 &...
- 981
0
votes
2
answers
97
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Optimization algorithms for Kronecker approximation of high-dimensional covariance matrices
I'm working with a high-dimensional covariance matrix and exploring Kronecker product approximations to make it computationally manageable.
Here's the setup:
I have a graph $G$ represented by a $D\...
- 1
0
votes
1
answer
236
views
Solving a 0-1 quadratic matrix inequality
I am working on a binary optimization problem. So far I have derived the following constraint functions.
\begin{align}
\begin{bmatrix} \left( P + \sum_{i=1}^n (\sum_{j=1}^n x_{i, j} \alpha_j) e_i e_i^...
- 11
1
vote
0
answers
46
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Intuition for proximal point method using L2 regularization
To minimize a function $f$, the proximal point method is defined as
$$x_{k+1} := \operatorname*{argmin}_x f(x) + \frac{1}{2\eta}\|x - x_k\|^2.$$
What's the intuition for why we want to use L2 ...
0
votes
0
answers
40
views
Iterating partially-unconstrained optimization with projection
Let $f:H\to \mathbb{R}$ be a strictly convex Fréchet differentiable, coercive function on a separable Hilbert space $H$ and let $C_1,C_2\subseteq H$ be closed and convex.
I want to optimize
$$
\tag{(A)...
- 5,405
1
vote
0
answers
37
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When does an optimal input sequence for a discrete-time system exist?
Suppose an LTI discrete-time system is given by the equations
$$
x_{k+1} = Ax_k + Bu_k,\\
y_{k} = Cx_k + Du_k
$$
with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\...
1
vote
0
answers
32
views
Variants of cutting plane method for convex optimization
The cutting plane approach in convex optimization is a general recipe for minimizing a convex function. The argument relies on the fact that using the gradient vector, we can cut the feasible set into ...
- 139
0
votes
0
answers
45
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Gradient-based optimization of $n$ functions
I appreciate the willingness of everyone to assist me in advance.
I am faced with a set of $n$ distinct convex optimization problems, each defined as follows:
\begin{equation}
\max\limits_{x \in \...
- 1
2
votes
1
answer
102
views
Norm bound in simultaneous stability to semidefinite program
In the context of robust control, I remember hearing that the two following problems are equivalent.
Find $P \succ 0$, such that $A P + P A^{\top} \prec 0$ for all $A \in \mathscr{A}$ where $$\...
- 275
0
votes
0
answers
45
views
Generalized envelope theorems
I'm looking for references for two generalizations of Danskin/envelope-type theorems for convex optimization. The first is for when the parameters are functions on a space rather than numbers. A ...
- 4,217
2
votes
0
answers
119
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Seeking insights on bounded set positive solutions for a set of linear systems in $\mathbb{R}^n$
Before delving into my query, I'd like to provide some context. Consider a continuous function $f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$ and a compact set $\mathcal{B}\subset \mathbb{R}^{k}$ (...
3
votes
1
answer
255
views
Is this constraint convex?
I have an optimization problem where the following constraint causes DCP Rule Error.
$$e^{x_n} \leq B \log _2\left(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\...
- 31
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
$$\max_{\lambda\in\mathbb{R}^m} g(...
- 275
3
votes
1
answer
370
views
Convex optimization without Slater's condition
In nearly all convex optimization methods that I read about, it is assumed that the problem satisfies Slater's condition, that is, there is a point that strictly satisfies all constraints (the ...
- 3,549
2
votes
1
answer
159
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Conic hull of a rectangle
I have a simple question that appeared in research: For a rectangle $S :=[a_1,b_1] \times[a_2,b_2] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n$. Let $p_0 = (a_1,a_2,\dots,a_n)$, and define $p_i ...
- 275
1
vote
1
answer
84
views
optimization over moving domains
Let $A, B$ be Banach spaces, and for any $a\in A$, $B_a\in B$ is a measurable subset. Consider the following optimization problem:
$$L(a)=\inf_{b\in B_a}\ell(b),$$
where $\ell(b)$ is a infinite-times ...
- 87
1
vote
0
answers
94
views
Linear Program Optimal Value
If $f(A,b,c)$ is the optimal value of a linear program
$\min c.x$
subject to $A.x \leq b ; x \geq 0.$
Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
1
vote
0
answers
70
views
LICQ vs MFCQ who is stronger [closed]
I want to ask you which constraint is stronger: MFCQ or LICQ.
- 11
0
votes
0
answers
201
views
How expensive is a proximal operation?
In optimization, a proximal step is usually considered as cheap as a gradient step. I'm quite confused by this. Why are people taking this convention at all?
A typical proximal step indeed incurs a ...
- 31
0
votes
0
answers
55
views
Relationship of optimal solutions between the total function and the sub function
This is an unconstrained convex optimization problem. Let $\mathcal{N}=\left\{1,\ldots,n\right\}$, $2\leq n<\infty$. Suppose there are many strongly convex functions $f_i(x)$, where $x\in\mathbb{R}^...
- 1
2
votes
1
answer
183
views
Exponential optimization problem
\begin{eqnarray}
\arg\max_{k}\sum_{i=1}^{p}\sum_{j=1}^{p}\exp\left(-{\frac{\left(X(i,j)-{U_k}(i,j)\right)^2}{2}}\right),\:\: k=0,\dots,p
\end{eqnarray}
where $X$ and $U_k$ are the $p\times p$ matrices,...
- 21
0
votes
1
answer
103
views
Constrained linear optimization problem on $C^1$
I am dealing with a problem of the form ($a<b$)
$$
\displaystyle \max_{v \in C^1([a, b])} \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \...
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
1
vote
2
answers
70
views
Monotonicity of kernel matrices with respect to hyperparameters
Let $\mathcal{X}$ be some nice space, let $\Phi$ be some ordered space, and let $K :\mathcal{X} \times \mathcal{X} \times \Phi \to \mathbf{R}$ be a positive-semidefinite kernel indexed by a ...
- 801
0
votes
1
answer
147
views
Is there a redundant constraint in linear programming? [closed]
From wikipedia:
But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice).
(In order to do that, ...
- 3
2
votes
2
answers
293
views
Optimal transport: the existence of an optimal pair of $c$-conjugate functions
$\newcommand{\diff}{ \, \mathrm d}$
Let
$X,Y$ be Polish spaces,
$\mathcal C_b(X)$ the space of all real-valued bounded continuous functions on $X$,
$\mathcal P(X)$ the space of Borel probability ...
- 835
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
0
votes
0
answers
166
views
Literature request: proving or disproving convexity of the optimal value function of semidefinite program (SDP) or convex optimization in general
Suppose I have a function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined as the following parametric optimization problem:
$$f(p) = \inf_xf_0(x) \quad \text{subject to } \quad G(x,p)\leq 0,$$
where ...
0
votes
0
answers
156
views
Optimal solution of complex optimization problem
Let $Q(x)=a(x)e^{jb(x)}$ be a complex function of $x$. We want to approximate this function with $R(x)=\alpha e^{jx\beta}$ such that
\begin{align}
\text{arg}\min_{\alpha,\beta} \int_{-\frac{A}{2}}^{\...
- 287
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
2
votes
0
answers
79
views
Convex optimization over compact sets defined as Aumann set-valued integrals
Let $(X,P)$ be a probability measure space. Let $K$ be a convex compact subset of $\mathbb R^d$ and let $F:X \to 2^{K}$ be a set-valued map. Assume that $F$ is:
closed (i.e $F(x)$ is closed for ...
- 6,853
1
vote
2
answers
270
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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
0
answers
29
views
Non-differentiability of the set of optima of certain optimization problems
Let $X \subseteq \mathbb{R}^n$ be compact and say the function $f \colon X \to \mathbb{R}$ is locally Lipschitz continuous. Say $\mathcal{X}$ is the set of all solutions of the following optimization ...
- 213
5
votes
0
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259
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Does the refined Slater's condition hold also in the infinite-dimensional case?
Let $X$ be an infinite-dimensional Banach space. I have the following optimization problem.
$$\begin{array}{ll} \underset{x \in X}{\text{minimize}} & f(x)\\ \text{subject to} & g_1(x) \leq 0\\ ...
- 51
1
vote
0
answers
73
views
Principal component analysis with boundedness constraints
Let $A$ be an $m\times n$ matrix with entries in $F$ ($F=\mathbb{R}$ or $F=\mathbb{C}$).
It is well-known that $A$ has decompositions of the form
$$\displaystyle A = \sum_{k=1}^r\lambda_k\hspace{2mm} ...
- 2,605
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
0
votes
0
answers
124
views
The best unitary matrices that approximate a matrix product
Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...
- 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
0
votes
0
answers
159
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Double summation of matrices as constraints in convex optimization in CVX
I want to implement the following optimization problem from the following paper Randomized Gossip Algorithms, Page 10 Eq 53:
\begin{align}
\text{minimize} &\qquad s\\
\text{subject to} & \...
1
vote
0
answers
157
views
Is $x\cdot f(x)$ quasiconvex for $x>0$, if $f(x)$ is monotonically decreasing, convex, and positive? [closed]
Original question
Given $f(x):\mathbb{R}^+\to\mathbb{R}^+$, which is monotonically decreasing and convex. Then define a function $g(x) = xf(x)$, I am wondering whether $g(x)$ is quasiconvex for $x>...
- 41
2
votes
0
answers
56
views
A variant of the elliptope relaxation
Given a p.s.d. matrix $A$, one may want to find:
$$
\max_x x^t A x \mbox{ such that } x \mbox{ has entries }1 \mbox{ or } {-1}.
$$
This hard problem has a well known relaxation based on the so called ...
- 2,772
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
0
votes
0
answers
127
views
On least-squares with positive semidefinite constraints
Given real symmetric matrix $\mathbf{R} \in \mathbb{S}^{n\times n}$ and matrices $\mathbf{X}_n, \mathbf{X}_{n-1} \in \mathbb{R}^{n \times m}$,
$$\begin{array}{ll} \underset{\mathbf{A} \in \mathbb{R}^{...
3
votes
0
answers
97
views
Projection onto level set of convex functional
Fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $F:L^2_{\mathbb{P}}(\mathcal{F})\rightarrow (-\infty,\infty]$ be bounded-blow, convex, lower semi-continuous, and not identically ...
- 5,405
0
votes
0
answers
92
views
Optimization problem where the objective function returns a function instead of a real number
As we all know, a classic optimization problem can be represented in the following way:
Given: a function $f: A \rightarrow \mathbb{R}$ from some set $A$ to the real numbers
Sought: an element $x_0 ∈ ...
- 141
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