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0
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0answers
136 views

Choose a right optimization method

I have a question related to optimization method selection. Suppose now I have some observation data (1D) as well as the underlying function model. For example, I use the following codes to generate ...
1
vote
1answer
151 views

Check Kuhn-Tucker conditions

How to check if $(0,1)$ point is the solution of this optimization problem using Kuhn-Tucker Theorem. Find the min of $e^{x_1-x_2}-x_1-x_2$ where $x_1+x_2\le1,\ x_1\ge 0,\ x_2\ge0$ I am thinking ...
1
vote
1answer
58 views

Find a decoupled explicit formula for a minimizer

Consider the energy $F(u,v) = \int^1_0((\frac{1}{4}(u')^2+(v')^2 +\frac{1}{2}(u-v+1)^2)dx$ for $C^1$ functions u and v on the interval (0,1) that satisfy the boundary conditions ...
4
votes
2answers
158 views

Finding an explicit expression for a minimizer

Suppose $f$ is a continuous function on the interval (0,1). We consider the energy functional $F(u) = \int^1_0\frac{1}{2}((u')^2+u^2)\,dx - \int^1_0fu\,dx$ which is well defined for continuously ...
3
votes
0answers
245 views

Optimizing non linear programs of two variables

The scenario is; We've got $n$ stationary 360$^{\circ}$ sensors in an confined area (each sensor is located at some arbitrary $\left(x,y\right) = \left(x_{n},y_{n}\right)$), once a unit $t$ enters ...
0
votes
1answer
51 views

non linear equations solving methods?

I need to find $l_{2}$ and $\theta$ numerically by solving below equations. How could I do that? At least do i have some iterative way of finding those two unknowns. All others parameters are ...
0
votes
1answer
242 views

Convex Functions: Property Proof

Let $f\colon S\to \mathbb R$ be a $C^1$ function on a convex domain $S \subseteq \mathbb R^n$. Show that if $f$ is convex then $(\nabla f(x) - \nabla f(y)) \cdot (x-y) \ge 0$ for all $x,y \in S$. ...
2
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1answer
38 views

Smooth Reformulation of NonSmooth Constraints

If I have something like : \begin{align} \min_x \max_i f_i(x) \end{align} I can reformulate this nonsmooth formulation as: $$\min_x z$$ $$z\geq f_i(x)$$ and I have a smooth formulation of the problem. ...
0
votes
0answers
182 views

Matlab quadprog lower bound constraint ignored?

I am trying to solve an optimization problem with matlabs quadprog function. I have set up the problem to solve only equality constraints with lower bounds of 0. ...
3
votes
2answers
139 views

Upper bound for Maximization problem

I have an optimization problem of the form Max $x_1+x_2+x_3+\cdots+x_n$ subject to $x_0^2+x_1^2+x_2^2+\cdots+x_n^2+x_{12}^2+x_{13}^2+x_{14}^2+ \cdots+x_{1n}^2+x_{23}^2 + \cdots +x_{2n}^2+ \cdots ...
1
vote
1answer
177 views

Convex Functions: Proofs

Let $f$ be a monotone nondecreasing function of a single variable which is also convex. Let $g$ be a convex function defined on a convex set $G$. Is it true that the composition of these functions ...
1
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0answers
64 views

Non-linear least squares, problem with shifts

There is a set of coordinates $P=\{P_i\}$. $P_i=[x_i,y_i]$ and a set of coordinates $Q=\{Q_i\}$, $Q_i=[X_i, Y_i]$, where $Q_i$ coordinates are given by the following non-linear functions $$X = f ...
0
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1answer
466 views

is nonlinear least square a non convex optimization?

linear least-squares are convex optimization. Are nonlinear least squares also convex optimization? Can someone please give some simple examples?
1
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0answers
47 views

Implementing Non Linear Optimization

I am trying to calculate model free implied volatility $\sigma_{\mathrm{MF}}$ for a relative performance index using the following method: $$ \sigma_{\mathrm{MF}}^2=2\sum_{i} ...
2
votes
1answer
149 views

non linear optimization

How to solve this optimization problem using matlab or some other tool. I know that, this is a convex problem with non-linear constraint $\rho\geq \rho_{min}$ , so i have tried many times it in ...
2
votes
1answer
115 views

Control on Conformal map

Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
3
votes
1answer
202 views

Optimization problem with ratio objective

I need to solve the following optimization problem $$ \text{maximize} \quad \frac{(a^T x)^2}{x^TBx+c^T|x|} \quad \text{subject to} \quad ||x||_1=1 \quad (\text{or alternatively} \quad c^T|x|=1), $$ ...
1
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1answer
471 views

Constrained Optimization - Lagrange Multipliers (Example)

Let $f(x, y, z) = xyz$ $h1(x, y, z) = x + y + z − 4$5 and $h2(x, y, z) = 2x − y$. Goal: Minimize $f(x, y, z)$ subject to $h1(x, y, z) = 0$ and $h2(x, y, z) = 0$. First part: Show that every ...
2
votes
1answer
516 views

Prove every local minimum is a global minimum

Let $Q\in\mathbb{R^{dxd}}$ and $A\in\mathbb{R^{d'xd}}$ be two matrixes and $b\in\mathbb{R^d}$, $c\in\mathbb{R^{d'}}$. Suppose $d'\lt d $. For $x\in\mathbb{R^d}$. Minimize $$f(x)= ...
5
votes
1answer
991 views

The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
4
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0answers
115 views

Why Compactness is Necessary at Minimax Theorem

According to Von Neumann's minimax theorem, I have $$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} \max_{x\in X} f(x,y)$$ for some compact sets $X$ and $Y$ and a convex (in $y$), concave (in $x$) ...
2
votes
1answer
868 views

Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by ...
3
votes
1answer
78 views

$P_{1c} = AP$ , $P_{2c} = BP$. How to find $P$? (being that $A$ and $B$ are $3\times 4$ matrices and $P$ is a $4\times 1$ vector)

This problem arose in my stereo vision project. $$ P_{1c} = A*P $$ $$ P_{2c} = B*P $$ where: $P_{1c}$ and $P_{2c}$ are $3\times1$ vectors, $A$ and $B$ are $3 \times 4$ matrices and $P$ is a ...
4
votes
4answers
385 views

Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).

To approximate a function $G$ over the interval $[0,1]$ by a polynomial $P$ of degree $n$ (or less), we minimize the function $f:R^{n+1} \to R$ given by $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$, where ...
2
votes
1answer
100 views

Nonlinear optimization question

For (x,y) in $\mathbb R^2$, consider f(x,y) = $x^2 -2xy + \frac{4}{3}y^2 - 4y$ Find the local minimum of f. Is it a strict local minimum? Compute the $\lim\limits_{|(x,y)|\to \infty}$ f(x,y) to decide ...
1
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0answers
43 views

Deconvoluting linear combinations of unknown distributions

Summary I am trying to deconvolute the distribution $T(x)$ of a population's $x$ parameter into sub-distributions ($P(x)$, $Q(x)$, $R(x)$ ...), of which I don't know the form (only that they have ...
1
vote
1answer
53 views

Reformulation of BQP to SDP

I run into the following reading some optimization papars: $$\min_x x^TAx $$ where $x\in\{-1,1\}^n$ and $A\in S_n$, Is equivalent to $$ \min <X,A>$$ s.t $diag(X) = (1,1,...,1)\;\; rank(X) = 1$. ...
1
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0answers
15 views

Existence of solutions to a Specific kind of non-linear system of equations with rational variables

Consider a system of the form $c_{11}a_1b_{1} + c_{12}a_2b_{2} + \cdots c_{1n}a_nb_{n} = q_1$ $c_{21}a_1b_{1} + c_{22}a_2b_{2} + \cdots c_{2n}a_nb_{n} = q_2$ $\vdots$ $c_{m1}a_1b_{1} + ...
4
votes
1answer
159 views

Kuhn-Tucker condition is not satisfied

Show that the solution to finding minimum of $f(x)=-x_{1}$ With conditions $-\sin(x_{1})+x_{2} \leq 0$ $x_{1}-x_{2} \leq 0$ is point $(0,0)$, but the Kuhn-Tucker condition is not satisfied in this ...
1
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0answers
49 views

why is it important to have $\max_x \min_y f(x,y)=\min_y \max_x f(x,y)$?

I am currently trying to understand the minimax theorem of Von Neumann and the improved versions of this theorem. At any case we have the property $$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} ...
0
votes
1answer
109 views

Minimization to Maximization doubt in SVM

I came across a lecture on Support Vector Machines and in the lecture they converted a maximization problem into a minimization problem. I am wondering how it was done... $ Max \frac {1}{||x||} $ ...
0
votes
2answers
145 views

Minimize $z=2x_1+3x_2-x_1^2-2x_2^2$ subject to $x_1+3x_2\le 6$, $5x_1+2x_2\le 10$, and $x_1,x_2\ge 0$

How can we minimize $z=2x_1+3x_2-x_1^2-2x_2^2$ subject to $x_1+3x_2\le 6$, $5x_1+2x_2\le 10$, and $x_1,x_2\ge 0$? I need to know the steps to solve or at least the guidelines as I am really new to ...
1
vote
4answers
71 views

A non-linear maximisation

We know that $x+y=3$ where x and y are positive real numbers. How can one find the maximum value of $x^2y$? Is it $4,3\sqrt{2}, 9/4$ or $2$?
1
vote
1answer
166 views

Max function on a closed compact convex set.

Consider a closed convex compact subset $\mathbb{S}$ of $\mathbb{R}^N$ while we denote any of its point by $x=[x_1,x_2,\ldots,x_N]^T$. Define the function \begin{align} f(x)=max(x_1,x_2,\ldots,x_N) ...
0
votes
1answer
139 views

Convex optimization problem to quadratic programming problem

Briefly, have the following problem: \begin{equation} \sum_{i = 0}^n a_i \ (max [ F_i( \bar x ), 0 ] )^2 \rightarrow min, \\\\ s.t.\\\\ A \bar x \leq b \end{equation} where $ F( \bar x ) $ is a ...
1
vote
1answer
83 views

(system of) nonlinear equations and instability

I heard that a system of nonlinear equations is unstable. I am curious of how "instability" is defined, and why do nonlinear equations show instability? Edit: OK, so what about contexts in matrices ...
4
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2answers
312 views

Analog of Simplex Method for Quadratic Programming

It happened so I need to work with an algorithm for solving QP problems. The main issue here is that I can't find any references to this algorithms (at least no references in sources in english). ...
2
votes
1answer
66 views

Maximalization of a cubic puzzle

What is the maximal volume of a post package of length $L$, width $W$ and height $H$, subject to the following restrictions: $L+W+H \leq 90 $ $L \leq 60$, $W \leq 60$, $H \leq 60$ Intuitively I ...
2
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2answers
82 views

A Quadratic Problem (which looks very simple)

This arises as a part of my work. \begin{align} \min_{x^{H}x=1}~&x^{H}A_1x \\ subject~to~&x^{H}A_2x=0 \end{align} $A_1$ and $A_2$ are $N\times N$ hermitian matrices and $x$ is a unit norm ...
0
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1answer
43 views

Concave optimal value?

Let $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{m \times n}$. Consider a compact set $C \subset \mathbb{R}^n$. For all $x \in C$ define $$ f(x) := \min_{y \in \mathbb{R}^m} \{ x^\top A y ...
1
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1answer
142 views

Lagrange method - non-linear system of equations

I have to compute optimal parametres of truncated cone so that its Volume is fixed (lets say it is 1) and its surface is minimal using Lagrange method These are equations desribing my object: ...
0
votes
0answers
72 views

Max Quadratic Expression

Let $A \in \mathbb{R}^{n \times n}$, $A = A^\top$, $B \in \mathbb{R}^{m \times n}$, and $\mathcal{C} \subset \mathbb{R}^n$ be a compact, convex set. For $A$ not negative semidefinite, how to globally ...
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0answers
320 views

Significance of Rank of Jacobian

I have been struggling with this question. What is the significance of finding the rank of a Jacobian matrix of a function? I understand that the Rank of a matrix signifies the number of linearly ...
3
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1answer
419 views

Minimizing with Lagrange multipliers and Newton-Raphson

I am writing a program minimizing a real-valued non-linear function of around 90 real variables subject to around 30 non-linear constraints. I found handy explanation in CERN's Data Analysis ...
1
vote
1answer
427 views

A sufficient condition for a unique maximum of the product of two concave functions

Given two concave functions $f(x)$ and $g(x)$, what conditions in terms of these functions can ensure that $h(x)=f(x)g(x)$ have a unique maximizer on an interval $[a,b]$ for $a<b$?
0
votes
1answer
30 views

Second order least squares problem with one parameter

I want to fit a set of measured data $x_i$ and $y_i$ to the expression: $$ \beta^2 + \beta x_i = y_i $$ $\beta$ is my only free parameter. Although this is a really simple expression, the standard ...
2
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0answers
200 views

sufficient condition for KKT problems

For the Karush-Kuhn Tucker optimsation problem, Wikipedia notes that: "The necessary conditions are sufficient for optimality if the objective function f and the inequality constraints g_j are ...
0
votes
1answer
62 views

Finding the value which minimises all residuals

I have a series of observations, measurements made at various times $t$. I now need to determine the most likely value of $R$ (distance) using the model below. The guide says I should find the value ...
2
votes
1answer
134 views

Positive values for a set of quadratic forms of Hermitian Matrices. (To find a set of vectors in which a hermitian matrix is positive definite)

Assume all matrices I discuss about are $N \times N$ and the vectors conform with dimensions. Consider the following set of Quadratic inequalities where all the matrices $A_i$ are hermitian. ...
3
votes
1answer
380 views

Lagrangian Multipliers

I have a fundamental question about Lagrange multipliers. Here it is: I have a function to maximize with respect to a parameter say $\theta$, subject to two constraints. Lets assume that the first ...