# Questions tagged [nonlinear-optimization]

Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.

443
questions

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22 views

### Clarification of second order condition on orthogonal optimization

In the paper A Feasible Method for Optimization with Orthogonality Constraints, the arthors write:
where
I am confused about the notation $\mathcal{D}(\mathcal{D}\mathcal{F}(X))[Z]$.
My ...

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8 views

### Constrained maximin optimisation problem

Let
i) $\mu = [\mu_1,\mu_2,\mu_3]\in\mathbb{R}^3$, such that $\mu_2 > \mu_1$, $\mu_2 >\mu_3$ fixed,
ii) $\lambda = [\lambda_1,\lambda_2,\lambda_3] \in \mathbb{R}^3$ such that $\lambda_1 \geq \...

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45 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 = ...

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85 views

### A minimax optimization with expectation operator

Let $\mathbf{X}$ be an $m\times n$ random matrix with Gaussian i.i.d entries with zero mean and unit variance. How we can think about the following optimization
\begin{align}
\max_m\mathbb{E}_\mathbf{...

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51 views

### the subdifferential at points of differentiability in infinite dimensional space

Let $ f:X \longrightarrow (-\infty,\infty] $ that $X$ is infinite dimensional space and $f$ be a proper convex function and $ x\in int(dom(f))$.
Is it the case that: if $f$ is differentiable at $x$, ...

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53 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)...

**-1**

votes

**1**answer

37 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}_{\!...

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votes

**1**answer

125 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&...

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65 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 $\...

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44 views

### Recovering sparsity from $\ell^1$ relaxation

I've read that $\ell^1$ norm on $\mathbb{R}^n$ is the convex relaxation of the $\ell^0$ "norm":
$$\|x\|_0 \triangleq \sum_{i=1}^n I_{x_i \neq 0}.$$
In the regression setting, under what ...

**1**

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**2**answers

87 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)}...

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61 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, $...

**2**

votes

**0**answers

120 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 ...

**2**

votes

**1**answer

119 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 ...

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44 views

### Lagrange multiplier theorem for nonnegative integral functional (fix issue with infinite integral)

Let
$(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space;
$N\in\mathbb N$;
$p_i$ be a probability density on $(E,\mathcal E,\lambda)$ for $i\in\{1,\ldots,N\}$;
$w_i:E\to\mathbb R$ be $\mathcal ...

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votes

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46 views

### Find minimizer of nonnegative integral functional over a closed convex subset of $L^2$

Let
$(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space and $$\lambda f:=\int f\:{\rm d}\lambda$$ for $\lambda$-integrable $f:E\to\mathbb R$;
$p:E\to[0,\infty)$ be $\mathcal E$-measurable ...

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vote

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23 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))...

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20 views

### Help finding sufficient conditions for unique maximizer to constrained maximization problem

I am working on a paper, and I have run into a constrained maximization problem. I would like to find some sufficient conditions for the maximizer (particularly, p) to be unique. Given my limited ...

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35 views

### Computational complexity of optimization algorithms using random algorithm theory

A fundamental and undoubtedly much-studied problem is that of determining not only whether or not an optimization algorithm converges to its optimum but also how fast it converges (see a discussion on ...

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66 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(...

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21 views

### Optimization problem related to knot placement for parametric interpolation

The problem of knot placement addresses the question of how to choose the parameter intervals $\lbrace[t_i,t_{i+1}]\,|,\, 0=t_0 \leqq t_i\leqq t_{n-1}\leqq t_n=n\rbrace$ in way that renders the ...

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18 views

### Backstepping control of second order nonlinear system

$\dot{x_{1}}=x_{2}^2-3sin(x_{1})x_{2}$
$\dot{x_{2}}=x_{1}^3-3x_{2}cos(x_{1})+u^\frac{1}{2}$
Question: Using the backstepping method and Lyapunov function, design the controller $u$ that will make ...

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vote

**0**answers

34 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

27 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)...

**3**

votes

**1**answer

112 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 ...

**0**

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59 views

### Compute weights (analytically or numerically) so that they minimize an integral

I have a measurable function $f:E\to[0,\infty)$, measurable weights $w_1,\ldots,w_k:E\to[0,1]$ with $\sum_{i=1}^kw_i=1$, positive probability densities $q_1,\ldots,q_k:E\to[0,\infty)$ and measurable ...

**2**

votes

**1**answer

98 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, ...

**1**

vote

**1**answer

68 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) = \...

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106 views

### Compute which of a finite number of integrals is minimal

Let
$(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space;
$f:E\to[0,\infty)^3$ be a bounded Bochner integrable function on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ ...

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votes

**0**answers

75 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-...

**2**

votes

**1**answer

126 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} \...

**7**

votes

**2**answers

625 views

### How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?

Related question asked by me on Math SE a few days ago: How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?
A few days ago, somebody asked How to prove $ \mathrm{e}^x\left|\...

**0**

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**0**answers

23 views

### Reference request for normalized gradient descent

Can someone introduce a good article/textbook explaining variants of the gradient descent method? In particular, I am interested in the normalized gradient descent where one works with
$\frac{\nabla ...

**1**

vote

**1**answer

89 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})^{\...

**3**

votes

**1**answer

71 views

### Largest subset not spanning the span

Let $S=\{c_1,\dots,c_n\}$ be a set of vectors in $\mathbb{R}^M$. Is the below problem studied in literature?
$$\max\limits_{S'\subset S} \vert S' \vert $$
$$s.t. dim(span(S')) < dim(span(S))$$
...

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votes

**0**answers

46 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 ...

**4**

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**0**answers

198 views

### A conjecture about the barycenter of a polytope

Could someone help me with the following conjecture? Thanks a lot!
Suppose I have a polytope $\Delta$ in $\mathbb R^n (n\geq 2)$ with coordinates $(x_1,x_2,\cdots,x_n)$ defined by linear ...

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114 views

### Minimax optimization of diagonal entries of function of matrix

Let $\mathbf{A}$ and $\mathbf{U}$ be arbitrary complex $M\times N$ and $N\times M$ matrices, respectively. Let denote superscript $(\cdot)^{\dagger}$ and $(\cdot)^{\mathrm{H}}$ as pseudo-inverse and ...

**2**

votes

**0**answers

26 views

### Criterion for optimality in two-step optimization procedure

Fix $n\in \mathbb{N}$ with $n>1$, let $X$ be an infinite-dimensional topological vector space and suppose that one is given:
continuous functions $F_0,\dots,F_n:X\rightarrow [0,\infty)$ for which $...

**2**

votes

**0**answers

50 views

### Convergence of Quasi-Newton method with fixed derivative

Consider the Newton iteration
$x^{(k+1)} = x^{(k)} - DF( x^{(k)} )^{-1} \cdot F( x^{(k)} )$
to find a zero of a function $F : \mathbb R^k \rightarrow \mathbb R^k$. If we freeze the first derivative,...

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113 views

### Span of a nonlinear function

Fix vectors $x,y\in\mathbb{R}^d$ and a smooth function $\phi:\mathbb{R}\rightarrow \mathbb{R}$. Define $\phi^d: \mathbb{R}^d \rightarrow \mathbb{R}^d$ as applying $\phi$ entrywise (i.e. $\phi^d(x_1, ...

**1**

vote

**0**answers

94 views

### Showing existence of a solution to an underdetermined system of equations with non-negativity constraints

Let $K$ be a positive integer, let $p\in (0,1)$, and let $\{W(k,i),W^B(k,i), \varphi_k(i)\}_{1\leq i\leq k\leq K}$ be variables.
I need to prove that there exists a solution to the following system ...

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36 views

### Minimizing along independent directions, nonlinear programming

Good afternoon, I am studying the book Nonlinear Programming: Theory and Algorithms (by Mokhtar S. Bazaraa, Hanif D. Sherali, C. M.) particularly the Theorem $7.3.5$. I'm not sure I understand this ...

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57 views

### Can we numerically solve this saddle-point problem?

Let
$(E,\mathcal E,\lambda)$ be a measure space;
$f:E\to[0,\infty)^3$ be $\mathcal E$-measurable with $\|f\|\in\mathcal L^2(\lambda)$;
$\tilde p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ for some $\...

**0**

votes

**0**answers

53 views

### Numerically solve a specific saddle-point problem

Let $(\Omega,\mathcal E,\mu)$ be a probability space, $k\in\mathbb N$, $$W:=\left\{w:E\to[0,\infty)^k:\sum_{i=1}^kw_i=1\;\mu\text{-almost surely}\right\},$$ $G$ be a finite nonempty set and $a^{(g)}:E\...

**0**

votes

**1**answer

70 views

### Suggestions for infinite horizontal optimization

I have been looking at this question for a while without any progress.
Question. Maximize
$$ I[\eta] = \int_0^\infty e^{-s} \Big[\sin\big(\eta(s)\big) + \sin\big(\sqrt{2}\eta(s)\big)\Big]\;ds$$
...

**0**

votes

**0**answers

20 views

### $G1$ interpolating curves with symmetric slopes in ends of segments

given a set $\lbrace p_i| 1\le i \le n\rbrace =\lbrace(x_1,y_1),\,\cdots,\,(x_n,y_n)\rbrace$ of points , which method can be recommended to calculate a sequence of angles $\left(\varphi_1,\,\cdots,\,\...

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31 views

### Minimization of a nonlinear smooth integral functional

Let $(E,\mathcal E,\lambda),(E',\mathcal E',\lambda')$ be measure spaces, $I$ be a finite nonempty set, $p,q_i$ be probability densities on $(E,\mathcal E,\lambda)$, $\mu:=p\lambda$, $\varphi_i:E'\to ...

**0**

votes

**0**answers

84 views

### Numerical solution of a nonlinear saddle point problem in a Hilbert space

Let $(E,\mathcal E,\lambda),(E',\mathcal E',\lambda')$ be measure spaces, $I$ be a finite nonempty set, $p,q_i$ be positive probability densities on $(E,\mathcal E,\lambda)$ for $i\in I$, $\mu:=p\...

**3**

votes

**0**answers

236 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 ...