A non-linear optimization problem includes an objective function (to be minimized or maximized) and some number of equality and/or inequality constraints where the objective or some of the constraints are non-linear. Use this tag for questions related to the theory of solving such problems or for ...

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29 views

Can $\min f'x$ s.t. $(a'x - b)^2 \le d $ be written as a SOCP?

It does not appear to be significantly different from the form listed here: http://en.wikipedia.org/wiki/Second-order_cone_programming with (in article notation) $i = {1}$, $ A = a$, and $b$, $d$ as ...
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93 views

Least squares problem with orthonormality constraints

Given $y_1,\ldots,y_n\in \mathbb{R}$,$w\in \mathbb{R}^d$, and $x_1,\ldots x_n\in \mathbb{R}^D$, how do we solve the following optimization problem \begin{align} \min_A \sum_{i=1}^n (y_i-w^TA^Tx_i)^2\\ ...
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187 views

Scaling factor and weights in Unscented Transform (UKF)

I'm trying to implement the UKF for parameter estimation as described by Eric A. Wan and Rudolph van der Merwe in Chapter 7 of the Kalman Filtering and Neural Networks book: Free PDF I am confused by ...
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1answer
56 views

Optimization problem with a minimization sub-problem as a constraint

I have a problem, for predefined $x_0,z\in\mathbb{R}$, which looks like $$\min_{\alpha,x} \sum_{i=1}^n \alpha_i f_i(x_i,z) $$ subject to \begin{align} \sum_{i=1}^n \alpha_i &= 1 \\ \sum_{i=1}^n ...
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42 views

How to reformulate this Set covering problem?

I am trying to solve the following implementation of the set covering problem of a crew rostering problem. Here constraint (19), meant to create a 12-hour break between the different shifts taken by ...
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1answer
45 views

Implement $\max$ with a closed form expression?

I have 2 functions: $f(x)$ and $g(x)$. Both of them range in $[0,1]$. Is there some way to define a $h(x)$ that efficiently takes the greater one of $h(x)$ and $g(x)$, i.e. $h(x) = \max \{ f(x),g(x) ...
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1answer
69 views

Showing a function is concave

Given $F(\underline{x}) = Ax_1 + Bx_2 + \ln(a^2-(x_1^2+x^2_2))$ on $S=\{\underline{x}\in\mathbb{R}\mid x_1^2+x_2^2<a^2\}$ with $A,B,a\in\mathbb{R}$, show that $F$ is concave on $S$. Since we have ...
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68 views

Identifying saddle points of a constrained nonlinear function with three variables

I know that if the Hessian matrix of a multivariable function at a given stationary point has both positive and negative eigenvalues then that stationary point must be a saddle point. Does the same ...
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18 views

Comparing the hardness of optimizing two similar, but different expressions

Suppose we have binary variables $y_1, ..., y_n$. To make the representation simple, we show the concatenated vector as $\mathbf{y} = (y_1, ..., y_n)$. Consider the two following functions: $$ ...
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0answers
118 views

Solving nonlinear matrix inequality - transformation to LMI

I have a nonlinear matrix inequality problem where $A,B,C$ and $M$ are known and T is unknown and I would like to find $T$ that satisfies $\begin{bmatrix} T^T M T + A & B \\ B^T & ...
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82 views

Lagrange condition and second-order conditions

Given a function to minimize or maximize with equality and/or inequality constraints, I can use Lagrange multiplier and/or KKT to solve such problems. So I understand how it works. My problem is ...
4
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58 views

QR-Decomposition of matrix valued function

I already posted the following question on MO, but id did not raise much interest there. Maybe the title is too elementary to gain research interest. Suppose I have a matrix valued function $$ ...
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19 views

What is some prerequisite to study nonlinear programming?

What is some prerequisite to study nonlinear programming? I already know calculus and linear programming is two perquisite, what else?
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1answer
54 views

Gradient of Objective Function

I want to know how to calculate the gradient $\triangledown f\left ( \mathbf{x} \right )$ of this functions: $f\left ( \mathbf{x} \right )=\left | \mathbf{a}^{H}\mathbf{x} \right |^{2}$, ...
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2answers
118 views

Optimization of Product of Different Objective functions (Ex.: Maximize The Product of projections of a complex vector)

Suppose We have this optimization problem which is convex $\mathbf{x}={\arg}\: \underset{\mathbf{x}}\max f_{i}\left (\mathbf{x} \right )$ But the product of different objective function is ...
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0answers
43 views

Inequality optimization, KKT condition.

So we have the problem: maximize $x^2+y^2$ subject to $x^2-y \leq3$ and $y\leq 1$. And I sorted out the KKT conditions for the problem (is here where the problem is?): $2x=\lambda _12x$, ...
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117 views

Find min of $\frac{\left( \sum kx_k \right) \left( \sum x^2_k \right)}{\left( \sum x_k \right)^3}$

With $n \ge 2$ and $x_1,\ x_2,\ \dots,\ x_n > 0$. Find the minimum of: $$ M = \frac{(x_1 + 2 x_2 + ...+ nx_n)( x^2_1 + x^2_2 +...+x^2_n)} {\left( x_1 + x_2 +...+ x_n \right)^3}$$ For specific ...
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3answers
251 views

Minimizing sum of squared distances from point to spheres

Given some spheres with known radius and known origin in three dimensional space, I want to find the point P that lies "closest" to all these spheres. The meassure of closeness, I guess, will be the ...
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1answer
21 views

Sets of feasible directions

I'm not exactly sure how the different points matter. I believe $p=[1,1,-1]^T, [2,-1,0]^T, [3,0,-1]^T, [0,3,-2]^T$ are all feasible directions.
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54 views

Maximizing the product of projections of a vector on another vectors

I want to get the $N\times1$ complex vector $\mathbf{x}$ which maximizes this real valued function $f=\mathbf{x}^{H}\left (\mathbf{a}_{1} \mathbf{a}_{1}^{H}\mathbf{x}\mathbf{x}^{H}\mathbf{a}_{2} ...
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1answer
40 views

Impact of constraints on convex optimization

Let me start by saying I know almost nothing about optimization so please bear with me. Basically, I am wondering whether it is possible to solve a problem with two constraints by solving the problem ...
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0answers
19 views

General 2D taylor surfaces from axial behaviour and discrete points

I have a problem as follows: I have a nonlinear function, f(x,y), for which I (numerically) know the axial behaviours, f(x,y0) and f(x0,y), where x0 and y0 are constants. I can calculate discrete ...
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46 views

Finding the maximum/minimum of a homogeneous function on $R^n$

Suppose that $f:R^n\to R$ is homogeneous. Also, suppose that the $argmin_xf(x)$ is non-empty. Is it true that if there exist $x^*\in R^n$ such that $f(x^*)=0$, then $x^*=argmin_xf(x)$?
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42 views

Nonlinear optimization in exponents

$$ max \pi = 4x_1^\frac{1}4x_2^\frac{1}3 - x_1 - x_2 $$ It is not difficult to determine that this function is concave and yields a global maximum at some point for the quantities $ x_1, x_2 >= 0 ...
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1answer
92 views

KKT point of a constrained optimization problem

Min$_{x}~x$ Subject to $x \geq 0$ For this problem, is $(x^{*}, \lambda^{*})=$$(0,0)$ a KKT point ? My try : I formulated corresponding Lagrangian and tried to find out the KKT point(s). ...
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1answer
47 views

Convex functions -> quasi-convex functions -> … can we weaken the assumptions?

First of all let me say that I'm new to optimization. I realized that quasi-convex functions share with convex functions some nice properties, so I wonder if we can push the weakening a little ...
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67 views

Unconstrained Nonlinear Nonconvex Optimization: LBFGS vs. Interior Point Methods?

I'm finding the literature on interior point methods somewhat inaccessible but I've found papers benchmarking different interior point methods for unconstrained nonlinear Nonconvex optimization. I ...
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1answer
46 views

least-square optimization with linearly depend solution $x$

What is the exact solution $x_{n \times 1}$ of the following constrained optimization problem \begin{align*} &\min \|A x - b\|^2 \\ s.t.& C x = 0 \end{align*} where $A$ is full column rank $m ...
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90 views

Minimizing a convex cost function

I'm reviewing basic techniques in optimization and I'm stuck on the following. We aim to minimize the cost function $$f(x_1,x_2) = \frac{1}{2n} \sum_{k=1}^n \left(\cos\left(\frac{\pi k}{n}\right) x_1 ...
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1answer
124 views

Can SVD help to solve (inequality) constrained least squares problem?

Consider the following minimization problem: $$ ||Q u - h^{o} ||^{2} \to min \;\;\; s.t. \; u \geq 0 $$ where $Q$ is $m \times n$ matrix and $u$ is $n$-dimensional vector and $h^{0}$ is ...
0
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1answer
24 views

Non-linear estimate parameter

I have one non-linear function that define $$E_x(a,b)=\int K_\sigma(y-x) \cdot(b-b. e^{-a\cdot f(y)} \,) dy$$ where $y$ is neighboor points of $x$; $f(y)$ is a function of $y$; and $a$ is constant. ...
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37 views

Noncontinuous subadditive function

Is there any non-continuous additive function $f(x+y)= f(x)+f(y)$ from $\mathbb R$ to $\mathbb R$?
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104 views

using lsqcurvefit to fit piece-wise linear

I would like to use this function to fit piece-wise linearly to a set of data. Namely, I want to fit them with several linear segments. Including other requirements, I would not want the segments ...
2
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0answers
131 views

Maximize a function subject to the constraint $x^2+y^2=R^2$

Please help me how to deal with maximization of function $$f(x,y)=1-e^{-\pi x}+e^{\pi x}\left[1-\cos(\pi y)+\sin(\pi y)\right]$$ subject to the constraint $x^2+y^2=R^2$. Using Lagrange ...
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99 views

transforming nonlinear matrix inequality to LMI

I faced some nonlinearity in my problems. I need to check a matrix inequality condition in order to check the feasibility of designed controller through a continuous design problem. My problem is that ...
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1answer
83 views

Why is this a quadratic programming problem?

I am sorry if this is a stupid question, I'm very new. How would I minimize the following objective? $\sum_{k=1}^p\| I_{k} - M_{k}A \|^2$ Each I and M are known. I am told I can use a quadratic ...
3
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1answer
47 views

How to find the minimal value of this function under such constraint

$f(x,y)=x-\sqrt{y-x^2}$ with $a<x<b$ and $x^2<y<c*x-d$. What I did is, first take partial derivative at $x$ and $y$ respectively, however, there is no critical point because fy is always ...
3
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2answers
162 views

Singular Values of Matrix as Optimization Problem

Assume that $A$ is a positive semidefinite symmetric matrix. It is known that $$\max_{||y||\leq1} \quad y^TAy$$ Has an analytical solution which is the maximum eigenvalue of $A$. This isn't hard ...
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43 views

Complex Non-liner First order ODE problem

Good day people I am modelling a "water bottle rocket" using basic Continuum Mechanics. I have found a equation describing the acceleration of the rocket. I need to integrate this function to find ...
2
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1answer
122 views

Extremum of functional of a complex function

consider functional $E$ defined by $$E[z]=\int F(x,z(x))dx$$ where $F$ is a complex-valued nonlinear function. How can we find the function $z(x)$ so that $$G=|E|^2=EE^*=\iint ...
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1answer
48 views

The most optimal way to solve this set of non-linear equations in high dimensions

So I have a series of non-linear equations which I wish to solve as fast as possible, to illustrate for the case of $n = 4$, I have the following equations: \begin{gather*} ...
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1answer
48 views

Is it possible to convert into linear programming problem

I have a problem of the form $$\sup_{x\in\Bbb{C}^n}\left\{\frac{\|Ax\|_\infty}{\|Bx\|_\infty}\right\}$$ where $A$, $B$ are matrices with different number of rows and $x$ is an $n$ dimensional vector. ...
2
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1answer
39 views

Conceptual Understanding of Non-Linear Optimization Problem

I'm in non-linear optimization, and I'm having trouble wrapping my head around what this problem is asking me for. If anyone could help with a conceptual explanation (not an answer!), it'd be greatly ...
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78 views

How to minimise an objective function which is not a direct function of the decision variable?

I have a problem with partitioning a water network by closing some pipes. I use some graph theory techniques to find some candidate pipes to close; but to select which pipes among them to close (my ...
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1answer
64 views

Prove that a multivariable function doesn't have global extremes

So my question is actually this. Say I have a function $F:\mathbb R^2\to\mathbb R$. If I find all the potential local extremes by finding the roots of the partial derivatives and I find that only one ...
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2answers
41 views

Constraints in optimization; redundant hardness?

This is not an accurate mathematical problem, and rather a philosophical and ambitious question. As far as I know, unconstrained problems are easier than constrained problems; right? This is mostly ...
3
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1answer
118 views

Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
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1answer
323 views

Converting Non-linear Programming Problem from Maximization to Minimization

I have a non-linear maximization problem and I want to convert it to be a minimization problem, can I do so by multiplying it by a negative sign, or is that wrong; and if that is wrong what should I ...
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2answers
290 views

What change of variables, if any, transforms this nonconvex problem into a convex one?

I'm looking for a convex reformulation, if any exists, of the following minimisation problem: Let $A$ be a symmetric, positive definite $n \times n$ matrix, and $b \in \mathbb{R}^n$. Minimise ...
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0answers
43 views

Effect of approximating a non-differentiable function on optimisation of minimisation

I am looking at a problem of constrained minimization, where the function to be minimized contains the Heaviside function, and as such is not twice continuously differentiable. My question is what ...