All Questions
Tagged with nonlinear-programming or nonlinear-optimization
639 questions
1
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0
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29
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Using Regula-Falsi to determine the solution to a non-linear system [closed]
Apologies, for this isn't a field or subject I know much about.
Regula Falsi (I believe some may know this as "double false position" or something like this) can be used trivially, of course,...
- 111
0
votes
1
answer
516
views
How do I get an analytical solution to this nonlinear equation?
I posted this question over on Math Stack Exchange (link), but have not received a response. I'm wondering if it's too complicated for that audience, so I'm posting it here in the hopes that someone ...
- 109
1
vote
0
answers
47
views
Prove that a polygon is convex over a circle
The problem
Let $C_A$ (resp. $C_B$) a circle of center $A = (x_A,0)$ (resp. $B = (x_B,0)$) and radius $r_A$ (resp. $r_B$).
For $k = 0,1,2,3,4$, let $D_k$ some points on $C_A$ with $D_0 = (x_0,0)$
Let $...
- 131
1
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0
answers
110
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Minimising kurtosis (non-convex). Can I use algebraic geometry or alternate methods to show uniqueness of a particular solution?
I consider a weighted sum of $n$ identically-distributed correlated random variables. The weights in the sum, $w_i$ for $i=1, 2,...,n$, satisfy $w_i>=0$ and $\sum_{i=1}^{n}w_i=1$. I am ...
- 173
2
votes
0
answers
101
views
Sparse signal recovery (nonlinear case)
Let $K \subset \mathbb{R}^n$, it may be that $K$ is "very thin" (e.g. $K$ is a $k$-dimensional affine subset of $\mathbb{R}^n$, with $k \ll n$). I'm interested in the case where $K$ is ...
- 981
1
vote
1
answer
195
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Metric / strong slope restriction of function on unit ball in $\mathbb R^m$
Diclaimer. I'm not sure this is the right venue for this question, but I'll give it a try
Definition [Strong / metric slope]. Given a complete metric space $(M,d)$ and a function $f:M \to (-\infty,+\...
- 6,853
5
votes
5
answers
623
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Elementary inhomogeneous inequality for three non-negative reals
I need the following estimate for something I am working on, but I don't immediately see how to establish it.
For $x, y, z \in \mathbb{R}_{\ge 0}$, show that
$$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz ...
- 543
7
votes
1
answer
371
views
An elementary inequality for three complex numbers
The following problem arose in asymptotic analysis of difference equations.
Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have
$$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)...
- 128k
0
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0
answers
36
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Optimizing upper and lower bounds
Let $L_i:X\rightarrow [0,\infty)$ be continuous (objective) functions defined on a metric space $X$ and suppose that
$$
L_1(x)\leq L_2(x)\leq L_3(x)\qquad (\forall x \in X).
$$
Here, I imagine that $...
- 5,405
2
votes
1
answer
245
views
Using Nelder-Mead to solve system of polynomial equations
I am trying to solve a system of $9$ polynomial equations in $9$ unknowns over the non-negative reals.
Since the equations are quite large and I would like to use VBA, I prefer an algorithm that ...
- 133
0
votes
0
answers
52
views
An optimization problem about number series
Given $m>0$, we want to minimize
$$
\sum_{k=1}^r a_k \log b_k
$$
for arbitrary increasing number series $a_k\geq 1$ and $b_k\geq 1$ satisfies
$$
\sum_{k=1}^{\infty} \frac{1}{a_k}=1
$$
and $r$ ...
- 1,503
7
votes
2
answers
632
views
Determining if a quadratic form is non-negative if variables are non-negative
Let $f(x_1,\dots,x_n) = \sum_{1 \le i \le j \le n} c_{i,j}x_ix_j$ be a homogeneous quadratic form. Is there a quick-ish way to determine whether $f(x_1,\dots,x_n) \ge 0$ for all $x_1,\dots,x_n \ge 0$?
...
- 1,355
1
vote
1
answer
331
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Dual problem with integrals
I am reading a paper where the author derives the following Lagrangian dual problem :
$\min_v \int_R \frac{1}{4} \frac{\beta^2}{v-2\|x\|}dx+v\;\;\;\text{s.t.}\;\;\;v\geq 2\|x\|\;\;\;\forall x \in R$
...
- 67
3
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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
0
votes
1
answer
153
views
Difference of two optimization problem's optimal value
Let we have two following optimization problems:
\begin{align}
\text{(P1)}\quad \alpha_1 = \max_{x_1,\ldots,x_M} &\quad \sum_{m=1}^{M}\log(1+f_m(x_1,\ldots,x_M))\\
\textrm{s.t.} &\quad \...
- 287
1
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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
$$
F(x)\geq k\|x\|...
- 5,405
2
votes
1
answer
148
views
Lower semi-continuity of length-dependent functional
Let $f:\mathbb{R}\rightarrow [0,\infty]$ be a lower semi-continuous function and define the functional
$$
\begin{aligned}
F_f:&\ell^1 \rightarrow [0,\infty]\\
(x_n)_{n=0}^{\infty} &\to \sum_{n=...
- 5,405
2
votes
0
answers
40
views
Numerical algorithms for geodesically convex optimization
I want to solve a minimization problem of the form
$\inf_{x \in M} f(x)$
where $M$ is a Hadamard manifold and $f$ is geodesically convex (but not differentiable). Since I know that in general a ...
user85365
0
votes
1
answer
194
views
Optimization problem involving matrix
I am struggling to solve an optimization problem of the following form:
$$\begin{array}{ll} \underset{A}{\text{maximize}} & \log \det (A) \\ \text{subject to} & a^T A^{-1} a \le b\end{array}$$
...
- 13
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
0
votes
0
answers
51
views
What transformation is required to find a unique solution of this problem instead of multiple solutions?
$$
\max\limits_{\mathbf{f},\ \|\mathbf f\|=1 } \log_2\left(\prod^K_{i=1} \ \frac{ \mathbf{f}^H {\mathbf E} (\mathbf{W}_i, \Theta, \tau_i) \mathbf{f}} { \mathbf{f}^H \mathbf{G}_i ( \mathbf{W}_i, \...
- 1
4
votes
1
answer
155
views
Is there a point in 6-dimensional space satisfying these polynomial inequalities?
I would like to know if there is a point $(a, b, p, q, x, y) \in [0,1]^6$ satisfying the following collection of inequalities.
$b \ge a$
$q \ge p$
$y \ge x$
$a \ge p \ge a^2$
$b \ge q \ge b^2$
$p \ge ...
- 543
3
votes
1
answer
413
views
Weird claims and conclusions in "Introduction to Shape Optimization"
I'm trying to understand the notions of Euler and Hadamard derivatives of shape functionals. All the lecture notes and papers on this topic that I've found seem to build up on the books Shapes and ...
- 167
4
votes
2
answers
209
views
If all points of a real function with positive values would be local minimum, can one say it is constant function?
During my studies I faced a function $f:\mathbb{R} \to \mathbb{R}^+ $ with the property: for all $x \in \mathbb{R} $ and all $y$ in open interval $(x-\frac{1}{f(x)} ,x+\frac{1}{f(x)}) $ we have $f(x) \...
- 179
2
votes
0
answers
46
views
Notion of distance between linear programs
Consider the linear programming problem
\begin{align}
\max_{x}&~c^Tx \\~s.t.~~a^Tx &\leq B~,~0\leq x_i \le1
\end{align}
where $c$ and $a$ are $n \times 1$ given non-negative vectors. $B$ is a ...
- 1,421
3
votes
0
answers
68
views
Convergence of iteration of a convex program
Let $\mathbf{V} \in \mathbb{R}_{+}^{n \times m}, \ \ \mathbf{E} \in \mathbb{R}_{+}^{n \times m}$, with $\mathbf{V} \mathbf{1}_{m} = \mathbf{1}_{n}$ and $\mathbf{E}^{T} \mathbf{1}_{n} = \mathbf{1}_{m}$...
- 53
1
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0
answers
163
views
Maximizing variance of bounded random variable through convex optimization
I am interested in maximizing the variance of a random variable $X$ supported on $[0,1]$. Formally,
$$\max_{P_X: X \in [0,1]} {\rm Var}(X),$$
where $P_X$ is a distribution of $X$. This question is ...
- 671
2
votes
2
answers
338
views
Of all probability matrix $P$ having stationary distribution $\pi$, find the one having smallest diagonal
I am requesting your help today trying to solve a somewhat odd problem. Is there a way to find through some numerical algorithm such as Newton's method the stochastic matrix $\boldsymbol{P}$ having ...
- 23
3
votes
1
answer
261
views
When is the optimum of an optimization problem affine in the constraint parameter?
While working on a variational problem I have reached to the following question:
Let $f:(0,\infty) \to [0,\infty)$ be a $C^1$ function satisfying $f(1)=0$. Suppose that $f(x)$ is strictly increasing ...
- 6,741
-1
votes
1
answer
768
views
How can I deploy a CG-Steihaug algorithm for trust region sub-problem solving?
I'm studying various optimization methods and on this occasion, I'm trying to tackle the Trust Region Problem by solving the sub-region problem with the Steihaug-CG algorithm in Python. I'm using the ...
1
vote
0
answers
82
views
Can this problem be rephrased as optimization on a manifold?
I have question. I have a Riemmanian manifold $\mathcal{M}$, like an $n$-dimensional regular surface in $\mathbb{R}^n$. And I have a smooth scalar field defined on this manifold $f:\mathcal{M} \to \...
- 115
3
votes
1
answer
228
views
Reference request: Variational techniques for complex "iterated" Lagrangians
I am interested in solving variational problems of the form
$$
\min_u \int \Big\{L(x,y,u(x,y)) + \phi\Big(\int J(z,y,u(z,y))\,dz\Big)\Big\} p(x,y)\,dx\,dy.
$$
for some known, smooth functions $L,J,\...
- 31
1
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0
answers
163
views
A real system of bilinear equations with $2n$ unknown and equations
I have the following system of $2n$ bilinear equations, for a square invertible matrix $A \in \mathbb{R}_{n \times n}$, and $2n$ unknowns organized in vectors $x,y \in \mathbb{R}^n$:
$$
diag(y) A x = ...
- 560
1
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0
answers
60
views
Maximum value of $\int (aF^2(x)g(x)+G^2(x)f(x))dx$ over all $f,g$ densities satisfying $\int F(x)g(x)dx=1/2$
I want to maximise $$I(f,g):=\int_{-\infty}^\infty (aF^2(x)g(x)+G^2(x)f(x))dx$$ where $a>0$ is a given constant, over all possible probability densities $f,g$ satisfying $$\int_{-\infty}^\infty F(x)...
- 319
-1
votes
1
answer
71
views
Compute the proximal of a mapping [closed]
Let $ f: \mathbb{R}\longrightarrow \mathbb{R}$: compute
proximal of following mapping
$$ f(x)= \sqrt {1-x^2} $$
for $ x \geq 0 $
I know that the proximal is given by
$$ \operatorname{prox}_{\!...
- 1
1
vote
1
answer
195
views
Is the minimum of a constraint optimization problem differentiable in the constraint parameter?
Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$, and suppose that $h(x)$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$.
Let $s&...
- 6,741
0
votes
0
answers
312
views
Convergence of heavy-ball method for non-convex optimization
The heavy-ball method (also called gradient descent with momentum) is commonly used in optimization. The update rule can be written as:
$$x_{t+1}=x_t-\eta\nabla f(x_t)-\beta (x_t-x_{t-1})$$
Suppose $\...
- 601
1
vote
1
answer
234
views
Log Fractional optimization problem
Let $\mathbf{x}$ be a vector of $N$ variables. Then, how can I solve the following optimization problem?
\begin{align}
\max_\mathbf{x}&\quad \sum_{n} \log(1+\frac{x_n}{\alpha+\sum_{m}\beta_m^{(n)}...
- 287
1
vote
0
answers
83
views
Gradient descent in $U(n)^r$
I have a function $f:U(n)^r\rightarrow \mathbb{R}$ which I would like to minimize. Here, $U(n)$ is the set of unitary matrices, and $r$ should be considered to be much bigger than $n$. For instance, $...
- 111
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
2
votes
1
answer
154
views
Is the optimum of this problem convex in the constraint parameter?
Let $f:\mathbb R^+ \to \mathbb R$ be a smooth function, satisfying $f(1)=0$, and suppose that
$|f|$ grows with the distance from $1$: $|f(x)|$ is strictly increasing when $x \ge 1$, and strictly ...
- 6,741
1
vote
0
answers
29
views
Gradient descent to estimate the ground truth pdf
I have a function $I_d(x)$ which defined over a plane. I could simulate the values of this function at different points. I have a ground truth probability density vector $p({\bf x})=(p_1(x),...,p_d(x))...
- 113
2
votes
0
answers
76
views
Minimizing $\int\lambda({\rm d}y)\frac{\left|g(y)-\frac{p(y)}c\lambda g\right|^2}{r((i,x),y)}$ with respect to discrete parameter $i$
Let $I\subseteq\mathbb N$ be finite and nonempty, $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space, $$\lambda f:=\int f\:{\rm d}\lambda$$ for $\lambda$-integrable $f:E\to\mathbb R$, $p:E\to(...
- 167
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 ...
1
vote
0
answers
81
views
Solve optimal control problem whose associated system is nonlinear
Solve the optimal control problem of the LQR kind
$$
\min_u \int_0^{+\infty} x_1^2+x_2^2+\gamma(u_1^2+u_2^2) \, dt \quad\text{such that}\quad \begin{cases}\dot x_1=\alpha(x_2-x_1)+u_1,& x_1(0)...
- 207
3
votes
1
answer
803
views
Maximum of sum of exponential function
Let $x_1,\dots,x_n$ be a set of given vectors in $\mathbb{R}_{+}^d$. Let $c_1,\dots,c_n$ be given positive constants. I am interested in finding the vectors $w_1,\dots,w_n$ in $\mathbb{R}_{+}^d$ that ...
- 1,421
1
vote
1
answer
791
views
Hardness of concave minimization problem
I have an optimization problem $\underset{x}{\min} ~ c(x) - k \cdot x$ where $c(x)$ is a non-decreasing concave function with $c(0) = 0$, $x \in C \subseteq \mathbb{R}^d_{\geq 0}$. By non-decreasing, ...
- 29
1
vote
1
answer
174
views
Solve a 2-dimensional optimal control problem via Riccati nonlinear equation
Consider the 2-dimensional optimal control problem of the LQR kind
$$
\min_u \int_0^\infty (x^T Q x + u^TRu) \, dt \quad\text{such that}\quad \begin{cases}\dot x(t) = Ax(t)+Bu(t) \\ x(0) = \...
- 207
0
votes
0
answers
279
views
What is the jacobian of an image lookup function?
I posted this question on Robotics Stack Exchange (link) but thought it could be relevant here as well.
I'm trying to solve a computer vision problem whereby I wish to use Levenberg–Marquardt non-...
- 101
2
votes
1
answer
188
views
Optimal function existence? what is it?
It's a problem abstracted from a real engineering project.
I want to find the best curve $y=y(x)$, $x \in [0,1]$: $y$ doesn't have to be a continuous function.
The constraint is
$$
L=\int_{0}^{1} \...
- 29