Questions tagged [nonlinear-optimization]

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

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32 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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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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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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1answer
172 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 (...
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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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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 ...
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23 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)...
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1answer
53 views

Hardness of concave minimization problem

I have an optimization problem $\underset{x}{\min} ~ c(x) - k \cdot x$ where $c(x)$ is a concave function with $c(0) = 0$ and $x \in \mathbb{R}^d_{\geq 0}$. Is this problem NP-Hard? I am aware that ...
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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 ...
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50 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 ...
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102 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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1answer
65 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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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-...
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1answer
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} \...
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560 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|\...
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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 ...
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1answer
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})^{\...
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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? $$\max\limits_{S'\subset S} \vert S' \vert $$ $$s.t. dim(span(S')) < dim(span(S))$$ ...
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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 ...
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1answer
150 views

A “nice” (but non-definite) quadratic programme

For integers $n\geq k>0$, let $f$ be the following quadratic form: $$f(x_1,\ldots,x_n)=\sum_{i=1}^n\sum_{j=0}^{k-1}x_ix_{i+j\bmod n}.$$ Is it true that the minimum of $f$ over the unit simplex is ...
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185 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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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 ...
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1answer
84 views

Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?

For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowlegde to solve it. Problem definition: Let $f(\xi) \in \...
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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 $...
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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,...
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2answers
313 views

Maximizing a function that is sum of gaussians

Let $\mathbf{x}_1,\dots,\mathbf{x}_n$ be given $n$ vectors in $\mathbb{R}^d$. Define the function \begin{align} \mathcal{K}(\mathbf{x},\mathbf{y})= \alpha\exp(-\frac{||\mathbf{x}-\mathbf{y}||^2}{2\...
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1answer
53 views

Reference request on computational schemes for $\inf_{x\in\Omega^n}\sup_{y\in\mathbb R^n}F(x,y)$

Let $\Omega\subset \mathbb R^d$ be compact, $\rho$ be a density function on $\Omega$ and $p_1,\ldots, p_n\in (0,1)$ be weights satisfying $\int_{\Omega}\rho(z)dz=1=\sum_{k=1}^n p_k$. We consider the ...
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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, ...
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2answers
160 views

Looking for a very particular kind of non-convex functions

I want some examples (hopefully parametric families!) of non-convex functions which satisfy the following properties simultaneously, It should be at least twice differentiable. It should have a ...
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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 ...
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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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53 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 $\...
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1answer
69 views

Suggestions for infinite horizontal optimization

I have been looking at this question for a while without any progress. Question. Maximize $$ I[\eta] = \int_0^\infty e^{-s} \Big[\sin\big(\eta(s)\big) + \sin\big(\sqrt{2}\eta(s)\big)\Big]\;ds$$ ...
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50 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\...
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1answer
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 ...
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1answer
261 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 ...
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1answer
153 views

How to determine the damping factor in Levenberg-Marquardt?

From the algorithm, we can see that it tries different damping factor until it gets a good one by the error. Is the damping factor related to the eigenvalues of the Hessian matrix?
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1answer
755 views

Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
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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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45 views

look for differentiable symmetric functions whose global minimizer has all distinct components

For symmetric functions, people ask Do symmetric problems have symmetric solutions?, e.g., [3] and [4]. The answer is no in general. However, solutions of symmetric problems often exhibit certain ...
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1answer
443 views

Difference between Chebyshev first and second degree iterative methods

Consider linear equation $Au = f$. We want to solve it with iterative method (assuming $A$ is good). First order iterative method is: $$ u^{k+1} = u^k - \alpha_{k+1}(Au^k - f), $$ The second degree ...
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30 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 ...
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1answer
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 ...
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1answer
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. ...
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81 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\...
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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 ...
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88 views

About martingales induced by iterative processes

Suppose I have a discrete stochastic process $\{ X_i \}_{i=1,\ldots..}$ defined as, $X_{i+1} = X_i - \eta \nabla f(X_i) + \sqrt{\eta} \xi_i$ where $f : \mathbb{R}^d \rightarrow \mathbb{R}$ and $\xi_i \...
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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 ...
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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,\...
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257 views

Computing Bohr Radii

The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as the radius $$R = \sup\limits_{0<r<1} \Bigl\{ r\ \Big|\sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D}, \text{ for all }f(z)=\...

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