For questions about optimization and optimization problems with non-linear restraints.

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2
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0answers
89 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 ...
1
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0answers
74 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 ...
1
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1answer
71 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 ...
0
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0answers
24 views

Non-linear, non-convex optimization problem

I would like to solve the following optimization problem: $$ \max\{\min_{k=1,2,...,m}\{R_d^k\}+\sum_{k=1}^m R_e^k\} \\\text{s.t.} \\ \mathrm{trace}(W_d^kW_d^{kT})=1 \\ \mathrm{trace}(W_e^kW_e^{kT})=1 ...
3
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1answer
45 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
124 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 ...
0
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0answers
86 views

Calculation of the set for the polar tangent cone?

I have the following theorem in my book. Assume that $\tilde{x}$ is a local minimum from a minimization problem and that f(.) is differentible at $\tilde{x}$ Let $T_X(\tilde{x})$ be the tangent cone ...
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0answers
33 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
91 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 ...
1
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1answer
40 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*} ...
0
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1answer
40 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. ...
0
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0answers
38 views

Combinations of Convex Functions

I'm looking at the following non-linear optimization theory problem: Let $\gamma$ be a monotone nondecreasing function of a single variable (that is, $\gamma(r) \le \gamma(r')$ for $r' > r$), ...
0
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0answers
38 views

Proofs involving closedness of compositions of mappings.

I've feel like I've gotten myself in over my head in a non-linear optimization course I seem to lack the mathematical maturity for(I'm an undergrad, I've taken the calc series, Intro to differential ...
0
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0answers
24 views

Implementing a projection with KL-divergence

I want to implement the following and I am looking for an easy/fast way to implement it(the programming language does not matter). Assume that $p(\mathbf{x})$ is a proper probability distribution and ...
2
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1answer
35 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 ...
0
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0answers
22 views

complexity of an optimization problem

Consider $n$ variables $x_1, \cdots, x_n$ with the constraint $\sum_{i=1}^n x_i=1$ and $x_i\geq 0$. I want to minimize $\vec{a}^T (I-\alpha A(\vec{x}))^{-1} \vec{b}$, where $\vec{a}$ and ...
1
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0answers
55 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 ...
1
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1answer
44 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 ...
0
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3answers
63 views

shortest distance between two points [duplicate]

I could not solve the following problem, Please help me, Let $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2)$ be two given points. find the third points $P_3=(x_3,y_3)$ such that $d_1=d_2$ is minimized, where ...
0
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0answers
47 views

is there any infinity norm bound to simplify this

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. ...
-1
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2answers
37 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
76 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 ...
0
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1answer
221 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 ...
2
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2answers
171 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 ...
1
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0answers
36 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 ...
1
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0answers
33 views

Representing a 2D function as a sum of rectangles of arbitrary shape and orientation

Suppose I am given a non-negative function $f(x,y)$ defined for $x \in [0,1]$ and $y \in [0,1]$. I'd like to represent this function as a weighted sum $w_i$ of a small number of rectangular apertures. ...
1
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0answers
591 views

Significant improvement when I use lsqnonlin function with wrong sized X

I was trying to solve a nonlinear least-square optimization problem using matlab function lsqnonlin with default algorithm trust-region-reflective. Let the optimization problem be "minimize ...
1
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2answers
412 views

How to find roots of a non linear multivariable equation using numerical methods

I started a course in linear algebra and numerical methods but I couldn't understand how can we numerically find roots of a nonlinear multivariable equation. f: Rn -> R Find f(x)=0 where x is ...
2
votes
1answer
58 views

Initialization of Limited-memory BFGS (using libLBFGS)

I am using the package libLBFGS in order to minimize an objective function, for which the first derivative (with respect to the optimization variable) is known and computable. I use the default ...
1
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1answer
45 views

How to prove the demicountinuity of nonlinear operators?

Define a nonlinear operator $\mathbf{J}(\mathbf{x}):~\mathbb{R}^3 \rightarrow \mathbb{R}^3$ as $$ \mathbf{J}(\mathbf{x}):= |\mathbf{x}|^{-\alpha}\mathbf{x},~0<\alpha<1. $$ How to prove that ...
2
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2answers
208 views

Are solutions to optimization problems with smooth, continuous, and strictly concave objective functions and linear constraints always unique?

As an example, if I have a minimzation problem where my objective function is represented by a sphere in n dimensions (one dimension per decision variable), and all my constraints are linear, then ...
1
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2answers
54 views

$f(x_1,x_2)=\frac{x_1^2}{x_2}$ quasiconvex and/or quasiconcave or nothing on $\mathcal R\times \mathcal R$?

Related to the 3.16e question in Boyd's book. It asks what is $f$ in $\mathcal R\times R_{++}$. I am not interested in it but related thing when the domain is larger. So $f(x_1,x_2)=\frac{x_1^2}{x_2}$ ...
1
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1answer
81 views

First-order necessary condition for relative minimum point

I'm studying linear and nonlinear programming and I came across with the following proposition : given $\rm x\in\Omega$ we are motivated to say that a vector $\mathbf d$ is a feasible direction at ...
1
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2answers
23 views

Show that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution

How can I show (preferably using the Bolzano-Weierstrass theorem), that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution? I can see that it is true, but how can I show it?
1
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1answer
40 views

A non-linear optimization problem

I have the following optimization problem on the variables $a_1, ..., a_n$: $$ minimize \frac{\sum_{k=1}^{n}\max(k\cdot a_{k},1)}{\sum_{k=1}^{n}a_{k}} $$ $$ such\ that\ \ 0\leq a_k\leq 1\ \ \ (k=1, ...
1
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0answers
62 views

Levenberg-Marquardt, QR decomposition

Could anybody explain, how the Levenberg-Marquardt method may be solved using the QR decomposition? I know a current solution ...
1
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0answers
239 views

How to solve nonlinear constrained optimization in Matlab?

I have to solve a nonlinear constrained function in matlab, and I am not familiar with it's commands. the problem is: minimize $E(b,c)$ constraints: $k1< c\sqrt{b}< k2 ; c/6>k3$ Note: E(b,c) ...
0
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0answers
320 views

How to linearize the product of two continuous variables in linear programming

I have a question when I deal with a linear programming model. The situation is that: I have some constraints in the model. All the constraints are linear, except some terms, which is the product of ...
2
votes
1answer
148 views

Prove or disprove the conjecture about the function below.

After thousands of numerical tests we stated the conjecture that their is exactly one local extremum of the function below. $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + ...
0
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5answers
130 views

How to find the minimum of the function?

How to find the minimum of the following function $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + {\rm e}^{-x_{i}\,w}} -y_{i}\right)^{2} $$ where $x_{i}, y_{i} \in \left(0, ...
3
votes
0answers
24 views

How does this polar function behave?

I came across this question in my textbook for Nonlinear Optimisation and I don't know what to do: Consider the function: $$ f(x_1,x_2)=(r-1)^2-\frac{1}{2}(r-1)^2\cos \left( \frac{1}{r-1}-\phi ...
1
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1answer
242 views

Definition of tangent cone in continuous optimization .

Looking at the definition of tangent cone in continuous optimization : If $M$ is a open subset of $\mathbb R^n$ $x \in M$, The tangent cone of $M$ at $x$ is defined by $$\mathbb T (M, x) = \big\{d ...
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0answers
59 views

Some problems with a proof of the Farkas Lemma

The following is a proof of the Farkas Lemma that is creating me quite some problems. [I presented the all proof simply to point out the notation used by the author.] My problem is with the last part ...
1
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1answer
114 views

Minimizing a linear function on a strictly convex set.

All the theorems that I know considering the uniqueness of a solution to a minimization/maximization problem requires the strict convexity/strict concavity of the objective function. But consider the ...
0
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2answers
33 views

Maximum of convex functions

how can i proof that: If $f_1, . . . , f_m$ are convex functions,than function $F(x) = \max(f_1(x), \dots , f_m(x))$ is convex? thanx for help.
1
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1answer
82 views

Third and higher order optimality conditions?

In the derivation of first and second order optimality criteria for a vector $X^*$ to be a local optimum to an unconstrained problem, we ignore the higher order terms of Taylor's expansion as we ...
0
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0answers
51 views

How do I find out if a critical point of a function is a maximum or a minimum?

If I've found the critical point of a function defined in some constraint (perhaps using Lagrange multipliers and the like); how do I find out if it's a relative/global maximum/minimum of a function ...
0
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0answers
24 views

How to check if steepest gradient method will converge?

So I have this function $ f(x,y) = x^4 - 2x^2 +x + 4y^2 $ and I want to know if the steepest gradient method will converge if I pick an arbitrary point and apply said method. My initial thought ...
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0answers
30 views

Defining a metric for non-convexity of functions

I have a non-convex function of two variables, $f(x,y)$. I drew it and realized that the function has a convex shape except in very few points where the non-convexity is very minor. I am looking for ...
3
votes
3answers
60 views

Minimization of $log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$?

I am trying to find the minimal value of the expression: $log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$ I think experience gives that the variables should be equal, if so then the minimal value is 6, but ...