Questions tagged [nonlinear-optimization]

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

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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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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)...
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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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1answer
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 ...
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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, $...
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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 ...
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1answer
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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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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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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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 ...
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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)...
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1answer
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 ...
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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 ...
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1answer
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, ...
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1answer
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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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-...
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1answer
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} \...
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2answers
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|\...
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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 ...
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1answer
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})^{\...
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1answer
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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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 ...
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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 ...
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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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0answers
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, ...
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0answers
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 $\...
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0answers
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\...
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1answer
70 views

Suggestions for infinite horizontal optimization

I have been looking at this question for a while without any progress. Question. Maximize $$ I[\eta] = \int_0^\infty e^{-s} \Big[\sin\big(\eta(s)\big) + \sin\big(\sqrt{2}\eta(s)\big)\Big]\;ds$$ ...
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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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0answers
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 ...
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0answers
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
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0answers
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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