# Questions tagged [nonlinear-optimization]

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

430
questions

**0**

votes

**0**answers

13 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 ...

**0**

votes

**0**answers

35 views

### Is there a closed form solution for this optimization problem?

The objective is
$$
\min_{\mathbf{x}_i\ge 0,\,\mathbf{1}^T \mathbf{x}_i=1} \sum_{i,j,i\,\ne j}^n\sqrt{\mathbf{x}_i^{T} \mathbf{x}_j}
$$
where the dimenson of $\mathbf{x}$ is $c$.
Obviously, if $n \le ...

**0**

votes

**0**answers

33 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 ...

**2**

votes

**0**answers

49 views

+50

### 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(...

**0**

votes

**0**answers

16 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 ...

**0**

votes

**0**answers

12 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 ...

**1**

vote

**0**answers

28 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

25 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)...

**2**

votes

**0**answers

45 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**

votes

**0**answers

51 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 ...

**1**

vote

**1**answer

83 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

66 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) = \...

**0**

votes

**0**answers

103 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 $p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ ...

**0**

votes

**0**answers

72 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

115 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

571 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**

votes

**0**answers

22 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

87 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

69 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?
$$\max\limits_{S'\subset S} \vert S' \vert $$
$$s.t. dim(span(S')) < dim(span(S))$$
...

**2**

votes

**0**answers

44 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**

votes

**0**answers

187 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 ...

**1**

vote

**0**answers

109 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 ...

**4**

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

45 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,...

**0**

votes

**0**answers

109 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

90 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 ...

**0**

votes

**0**answers

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 ...

**0**

votes

**0**answers

55 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

51 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

69 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,\,\...

**0**

votes

**0**answers

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

82 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 ...

**0**

votes

**1**answer

55 views

### Is there a general guideline for minimizing $\sup_{y\in H}F(\;\cdot\;,y)$?

Let $H$ be a $\mathbb R$-Hilbert space and $F:H^2\to\mathbb R$. Is there a general guideline for minimizing $\sup_{y\in H}F(\;\cdot\;,y)$?
Since the question is rather abstract, feel free to impose ...

**1**

vote

**0**answers

45 views

### Reduce the asymptotic variance for a class of Metropolis-Hasting estimates

I'm running the Metropolis-Hastings algorithm with state space $E$, target distribution $\mu=p\lambda$ and proposal kernel $Q$ to estimate $\mu(hf)$ for a fixed function $f:E\to[0,\infty)^3$ and a ...

**2**

votes

**0**answers

126 views

### Can we conclude $\sup_g\int f_1g\le\sup_g\int f_2g$ from $\int f_1\le\int f_2$ in this situation?

Disclaimer: Please bear with me, the question isn't as complicated as it looks like, but I wasn't able to find any simplification for which no counterexample comes to my find.
Let $(E,\mathcal E,\...

**0**

votes

**0**answers

31 views

### Solve a system of integral equations resulting from the Lagrange multiplier theorem

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 $(E,\mathcal E,\lambda)$, $\varphi_i:E'\to E$ be bijective ...

**1**

vote

**0**answers

49 views

### Maximize a smooth integral functional by pointwise maximization of the integrand

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 $(E,\mathcal E,\lambda)$, $\varphi_i:E'\to E$ be bijective ...

**1**

vote

**0**answers

75 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\...

**0**

votes

**0**answers

57 views

### How to maximum L1 norm problem?

I have met a problem these days.
\begin{equation}
\underset{\omega}{\max} \quad \Vert \text{diag}(\mathbf{h}^H)\mathbf{G}^H\mathbf{\omega}\Vert_1 \\
s.t.\quad\mathbf{\omega}^H\mathbf{G}\mathbf{G}^H\...

**1**

vote

**2**answers

162 views

### A four-variable maximization problem [closed]

We let function
\begin{equation}
\begin{aligned}
f(x_1,~x_2,~x_3,~x_4) ~&=~ \sqrt{(x_1+x_2)(x_1+x_3)(x_1+x_4)} \\
&+ \sqrt{(x_2+x_1)(x_2+x_3)(x_2+x_4)} \\
&+ \sqrt{(x_3+...

**1**

vote

**0**answers

85 views

### Show a Poincaré inequality for a Markov kernel and minimize the Poincaré constant

Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$ (see definitions ...

**2**

votes

**1**answer

142 views

### Is there a closed-form solution for this problem?

Let $A\in\mathbb{R}^{m\times n}$ be a full column rank matrix. Then there exists a left inverse $A^+$ of $A$. Let $w\in \mathbb{R}^n$ be a vector. Is there a closed-form solution for the following ...

**2**

votes

**1**answer

132 views

### prove $\log \left[ {\sum\limits_{i = 1}^M {{\varepsilon _i}{{\left[ {Q\left( {{a_i} + {b_i}\sqrt u } \right)} \right]}^2}} } \right]$ is convex

I am having difficulties to prove $\log \left[ {\sum\limits_{i = 1}^M {{\varepsilon _i}{{\left[ {Q\left( {{a_i} + {b_i}\sqrt u } \right)} \right]}^2}} } \right]$ is convex for non-negative a, b,u. ...

**0**

votes

**0**answers

40 views

### Is there a multiplier rule for this minimization problem?

Let $(E,\mathcal E)$ be a measurable space, $W\subseteq\left\{w:E\to\mathbb R\mid w\text{ is }\mathcal E\text{-measurable}\right\}$ be a Banach space, $k\in\mathbb N$ and $f:W^k\to[0,\infty)$. I'm ...

**1**

vote

**0**answers

86 views

### solving a non-linear Matrix equation

I am working on a problem of optimization of wireless sensor networks in order to achieve the optimal power allocation in the network. I find an equality matrix problem that can be written as ...

**2**

votes

**1**answer

173 views

### Understanding an argument involving the Sherrington-Kirkpatrick spin glass model

I have a question regarding an argument in
Aizenman, M.; Lebowitz, J. L.; Ruelle, D., Some rigorous results on the Sherrington-Kirkpatrick spin glass model., Commun. Math. Phys. 112, No. 1, 3-20 (...

**0**

votes

**0**answers

79 views

### Minimize an integral functional on a convex set

Let $$G(w):=\sum_{i\in I}w_i\;\;\;\text{for }L^2(\mu)^I.$$ I want to minimize $$F(w):=\sum_{i\in I}\sum_{j\in I}\int\lambda({\rm d}x)w_i(x)p(x)\int_{\left\{\:pq_j\:>\:0\:\right\}}\lambda({\rm d}y)\...

**0**

votes

**1**answer

48 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 ...