Tagged Questions

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

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0
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1answer
29 views

Non linear programming

Could you please help me in solving the problem posted below. A company uses a raw material to produce two types of products. When processed, each unit of raw material yields 2 units of product 1 and ...
0
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0answers
12 views

Linear independence of equality constraint gradients in constraint qualifications

I'm, trying to get an intuitive feel for the various constraint qualifications for KKT points. Most of them seem to rely on the linear independence of $\nabla g_i(x^*)$ where $g_i$ are the equality ...
-5
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0answers
25 views

problem on dynamical system [on hold]

Sir plz tell me what is self organised criticality?in terms of statistical mechanics
1
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0answers
18 views

Reduce degree of a high degree unconstrained binary term to quadratic unconstrained binary term

I'm working on a optimization project, in this project I have to convert higher order unconstrained binary polynomial to quadratic unconstrained binary polynomial. Can anyone give me a hint of how to ...
1
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0answers
47 views

Mixed-Integer Linear Programing : get the maximum constant associated to a non null variable

Does anyone know a way get the maximum constant associated to a non null variable using Mixed-Integer Linear Programing ? I would like to get the variable $a$ in this description : $$ i = 1,\ldots,m ...
1
vote
0answers
14 views

Can this specific Linear Program constraint be expressed? [duplicate]

Thanks for your time. I have a linear program and no idea how I could express a form of constraint and even if it's possible. Maybe someone here know a solution. A company assembly and sells a ...
1
vote
0answers
13 views

Line search Armijo, Wolfe, Strong Wolfe and Goldstein.

What are the articles (References) who proposed the line search of Armijo, Wolfe, Strong Wolfe, and Goldstein? Articles precursors of unidirectional searches?
0
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0answers
16 views

In the problem below, what is the right mix of drugs to maximize the expected revenue without exceeding R&D resources?

This is not a homework question. It's a question for a class I have yet to take that my friend gave me. I haven't been able to figure it out. Help is appreciated because it's driving me crazy. A ...
2
votes
0answers
22 views

Exercice the converge $r$-superlinearly

Give $x_0 \in \mathbb{R}^*$. Show that $\{x_k\} \subset \mathbb{R}$ converge $r$-superlinearly for $x^∗=0$, where $x_k$ is defined by $x_{k+1}=(1−\beta_k)x_k$ and $\beta_k=1−2^{-k}$ if $k=i^2$ for ...
1
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0answers
42 views

Proving/deciding concavity of a function of two variables

I would like to formally prove that the function $f(x,y) = \frac{(c+1)e^{-x}(xe^{x+y}+y)}{(c+2)(e^{x+y}-1)+e^y} $ is concave ($ c>2$ is a constant, and both $x,\, y \in \mathbf{R_+}$). Plots of ...
0
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0answers
17 views

gradient descent - cost reduces and then increases

I am optimizing a function using Gradient Descent. The learning rate is fixed. First for few iterations the cost decreases after that it starts increases. What is the reason for this?
0
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0answers
41 views

optimization of nasty expression with nice symmetry between expressions

Consider the function $\ f(x,y,z,\rho_a,\rho_b)=$ $ \log \left(1+ (x+ \rho_ay)^2 + \frac{(z+ \rho_by)^2}{1+(x+ \rho_by)^2} \right)+ \log\left(1+ (x+ \rho_by)^2 + \frac{(z+ \rho_ay)^2}{1+(x+ ...
4
votes
1answer
130 views

Lipschitz continuity of parametric optimizer

Consider the parametric optimal solution $x^{*}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ defined as $$ x^*( y ) := \arg\min_{x \in X } \ \ x^\top x + x^\top A y \\ \quad \qquad \text{subject to: } \ ...
1
vote
0answers
26 views

When result of max of min problem is equal to min of max problem

Let's assume there are two functions $f(x)$ and $g(x)$. I want to know when the optimal $x$ of max of min of $f(x)$ and $g(x)$ is not equal to optimal $x$ of min of max of $\frac{1}{f(x)}$ and ...
1
vote
2answers
43 views

Is the set of all projection matrices a convex set?

The set $\phi=\{P| P^2=P\}$ contains all projection matrix. Is this set $\phi$ convex?
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0answers
13 views

Sequential Quadratic Programming

I am new to optimization techniques. I have lograthmic function which I need to maximize. I need to ask if sequential quadratic programming can be used to solve logrithmic function taken as objective ...
0
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1answer
23 views

Solving coupled non-linear equations

I am struggling to understand what the following question requires me to do: I believe I need to differentiate implicitly, but am unsure how I show it cannot be done.
1
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0answers
18 views

Explain KKT conditions without reference to duality.

Is it possible to explain (not derive) KKT necessary conditions without reference to the concept of Lagrangian duality?
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0answers
24 views

Constrained non-linear optimization problem

For some background, this comes from a sample size allocation problem in statistics. I am trying to minimize the following function (a sum of three variances), and could use some help with direction ...
0
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0answers
29 views

Saddle point problem (KKT) with block-diagonal matrix

Consider the following saddle point problem originating from an interior-point method algorithm: $$ \begin{bmatrix}\mathbf{H} & \mathbf{A}^{T}\\ \mathbf{A} & \mathbf{0} ...
0
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0answers
15 views

Non linear Programming Problem

I am struggling with the following question: Solve the following programing problem: max $f(x_1,x_2)$= $ \sqrt{(x_1 + 1) (x_2+1)} $ subject to $x_2-(x_1-1)^2 \leq 0 $; $x_1+x_2 \leq 7 $; $x_1, x_2 ...
0
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0answers
17 views

Question regarding the solution of a nonlinear algebra equation

I want to prove for a nonlinear equation $x=f(x)$, the solution for this equation $x_0$ will maximize the first order of $f(x)$. i.e. $x_0=argmax(f'(x))$. Here are what we have $f(x)$ is a ...
0
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0answers
18 views

Separate a list of spheres into several lists, each contained in a sphere with a radius no larger than specified.

I have a list of arbitrary spheres, what I want to end up with is that list separated into a number of groups, where spheres in each group all fit into thier specific larger sphere. The limitation is, ...
0
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0answers
33 views

Fenchel dual vs Lagrange dual

Consider the Fenchel dual and the Lagrangian dual. Are these duals equivalent? In other words, is using one of the these duals (say for solving an optimization), would give the same answer as using ...
1
vote
1answer
33 views

Question regarding KKT conditions in optimization

Following is Proposition 3.3.7 in Bersekas' Nonlinear Programming. Let $x^*$ be the local minimum of the problem: $$\text{Minimize }\; f(x) $$ $$ \text{subject to: }\ h_j(x) = 0, ...
7
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0answers
67 views

Consider the problem minimize $f(x)= x^4 −1. $

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
1
vote
0answers
19 views

Solving many independent non-linear systems simultaneously

I'm working on solving lots of systems of nonlinear equations. Luckily, the non-linear equation is the same, but the parameters are different: $$ f(\vec{x}_0; c_0) = 0\\ f(\vec{x}_1; c_1) = 0\\ ...
0
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0answers
14 views

Quasiconcave condition for a power function

Let $f(x, y)= (ax^2+by^2)^n$ where $a, b, n$ are positive, $x, y\in \mathbb{R}$. What is the condition of $n$ so that $f(x, y)$ is a quasiconcave, and concave function? My idea is only calculate ...
2
votes
1answer
52 views

min : sum of L2 norm and squared-L2 norm.

Is there a closed-form solution of the following convex problem: $$\min_x \| x - u \| + C \| x - v \|^2$$ where $\| \cdot \|$ is the L2 norm.
1
vote
2answers
24 views

How to mathematically prove the optimality conditions for a univariate function?

Consider a univariate function $f(x)$. I know the graphical intuition behind why $f'(x)=0$ at the extrema of $f$. But how do you prove it mathematically? I start with the assumption of $x^*$ being a ...
1
vote
1answer
35 views

Why don't we go beyond the Hessian in multivariate optimization?

In univariate optimization, we perform the first derivative test to identify stationary points and the second derivative test to classify the stationary points as minima, maxima and inconclusive. When ...
1
vote
1answer
46 views

Constrained Optimization : Minimize sum of dot products

I am working on a problem to minimize sum of dot product. The problem can be stated as following. Given a matrix where each element is either 0 or 1. $$ \ A_{ij} = \{0,1\}; $$ with the constraint ...
2
votes
0answers
33 views

Constrained non-linear Optimization using Newton's method - Portfolio optimization

I want to solve following constrained optimization problem from portfolio optimization: The solution is supposed to be a modified risk parity portfolio: The optimization problem is: \begin{align} ...
0
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0answers
14 views

Nonlinear programming-separable programming

i have this function: $$5x_1x_2+4x_2x_3$$ and i need yo know if is separable or not I guees is not separable, because i can´t write the function in form: $$f_1(x_1)=x_1$$ and $$f_2(x_2)=x_2$$ ...
1
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0answers
91 views

How can I find the unit vector that minimizes the number of nonzero projections that a set of points has on it?

$\underset{\mathbf{w}}{\min} ~ \|\mathbf{X}^T\mathbf{w}\|_1~~~\text{subject to:}~ \|\mathbf{w}\|_2^2=1$ where $\mathbf{X}\in\mathbb{R}^{d\times m}$ is a set of $d$-dimensional points and $m>d$. ...
0
votes
1answer
33 views

Minimizing a non-convex rational function of two variables

I need to minimize the following function $$f(x,y)= \frac{a}{x}+\frac{bx}{y}+\frac{cy}{x}+dy+\frac{e}{y}$$ where $a,b,c,d$, and $e$ are positive constants, and $x$ and $y$ are both strictly positive. ...
2
votes
0answers
29 views

Constrained Quadratic Optimization(Reproducing Kernel)

I am attempting to use a constrained quadratic optimization to find the coefficients of a reproducing kernel. The problem is as follows: $y(t)=\sum_{i=0}^J\alpha_iK(t, t_i)$ $Q(\alpha)= ...
0
votes
2answers
18 views

Solving for gradient of Frobenius norm term

Let's first define a couple of variables: $A,B,C \in \mathbb{R}^{m \times n}, D \in \mathbb{R}^{n \times n}$, and $\mu$ is a scalar. Say I have an ADMM sub-problem that looks like this: $\arg ...
1
vote
1answer
134 views

How to solve this optimization problem with equality constraints?

I want to find $\delta_j$ in the following optimization problem. My variables are $\gamma_i$ and $\delta_j$ (all other symbols are known parameters). Assume $i\in\{1,\ldots,9\}$ and ...
0
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0answers
16 views

Non-linear optimization with unknown derivative and real numbers only

Background I'm trying to optimize a set of 7 parameters which are the core configuration of some external engine (specifically Solr parameters). I already have an optimization function which grades ...
0
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0answers
23 views

Non-convex constraint made cost

Consider the non-convex optimization problem $$ \min_{x \in X} \ f(x) \quad \text{s.t.:} \ \ g(x) \leq 0, \ h(x) = 0 $$ where $X \subset \mathbb{R}^{2n}$ is compact and convex, $f$ and $g$ are ...
1
vote
0answers
26 views

Maximizing the uniformity of density function subject to moment constraints

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
0
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0answers
33 views

Concave Quadratic Program

Let $X \subset \mathbb{R}^n$ be compact and convex. Consider $$ x^* := \arg\min_{x \in X} x^\top Q x + c^\top x $$ where $Q \prec 0$. I am wondering if there are cases where $x^*$ can be written as ...
1
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0answers
27 views

Hint for KKT Optimization problem

Can anyone help me with the following optimization problem please? I have to find the $\max f(c,y_1^1,\cdots,y_{N-1}^1,\cdots,y_1^M,\cdots,y_{N-1}^M)=c$ subject to the constraints ...
0
votes
1answer
69 views

Convex optimization approximation

Consider the optimization problem $\mathcal{P}_0$ $$ \min_{x \in \mathbb{R}^2} \left\| x-p \right\|^2 $$ $$ \text{sub. to: } \ A x \leq b, \ \ x_1^2 + x_2^2 = 1 $$ where $p \in \mathbb{R}^2$ is a ...
0
votes
1answer
52 views

Model $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty $ into standard form

I need to solve the following convex optimization problem: $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty$ s.t $x ...
1
vote
1answer
34 views

Minimize $\ell_1$ norm subject to $\ell_2$ constraint

I am trying to solve the following optimization problem: $$\min_{\|Px\|_2=1} \|x\|_1$$ I know it is non-convex. But some non-convex problems are still solvable. Update $P$ is 2x3. $x$ is a ...
3
votes
0answers
129 views

Finding optimal velocity profile using Dynamic Programming

This question has been asked on scicomp but I thought maybe it's more a mathematical problem of how Bellman's idea is to be applied here. The main problem for me is: How to introduce the time ...
2
votes
0answers
62 views

Levenberg's original article “A method for the solution of certain problems in least squares”

Does there exist any digital copy of the original article (or a transcript) K. Levenberg, A method for the solution of certain problems in least-squares, Quart. Appl. Math. 2 (1944): 164-168? It is ...
0
votes
0answers
30 views

Constrained optimization using a cutting plane on a tetrahedron

Consider the figure below where $(a,b,c,d)$ is a tetrahedron and $p=(1-t)a+tb$ is a point on the $ab$ segment. If $n_a$ and $n_b$ are two unit vectors associated with $a$ and $b$, respectively, then ...