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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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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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.
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38 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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39 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 ...
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74 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} ...
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Maximizing curious symmetric function from simple combinatorics

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). (This question has been posted at ...
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28 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 ...
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32 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, ...
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154 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 ...
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80 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, ...
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124 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) ...
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73 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\\ ...
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153 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.
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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 ...
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1answer
62 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 ...
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1answer
115 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 ...
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86 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} ...
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97 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 $m>d$. ...
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51 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. ...
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84 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)= ...
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2answers
64 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 ...
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1answer
163 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 ...
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23 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 ...
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31 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 ...
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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 ...
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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 ...
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92 views

Convex optimization approximation

Consider the optimization problem $\mathcal{P}_0$ \begin{equation*} \begin{aligned} & \underset{x\in \mathbb{R}^2}{\text{minimize}} & & \left\| x-p \right\|^2 \\ & \text{subject ...
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74 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 ...
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1answer
49 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 ...
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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 ...
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216 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 ...
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56 views

Evolutionary algorithm

Can someone provide me a good reference for the CMA-ES algorithm? I'm new in the world of optimization and just reading the author reference doesn't help me a lot. I know the basic idea of a genetic ...
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306 views

Maximizing “log det + log sum exp” function

I'm trying to find a numerical solution to the following optimization problem $$ \text{maximize } f(M) = \frac{1}{2} \log \det(M) + \log \sum_{i=1}^n \exp \left\{ - \frac{1}{2} x_i^T M x_i + a_i ...
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1answer
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Optimization problems on the circle

Consider the optimization problem $$ \min_{x \in \mathbb{R}^2} x^{\top} P x + q^{\top} x$$ subject to: $$ A x = b, \ x \in X, \ x_1^2 + x_2^2 = 1$$ where $X$ is compact and convex. Then ...
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1answer
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Analysing/Visualising shape of multi-variate function.

I have an unknown function $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ for which I'm determining a first order Taylor approximation through a non-linear optimization process in six variables (the ...
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1answer
58 views

project a point onto the intersection of surfaces

I have several non linear equations $g_i$ that represent surfaces $s_i$. Their intersection form the surface $S$. For example $s_1 : g_1(x_1,x_2,...,x_n)=c_1$ ... $s_n : g_m(x_1,x_2,...,x_n)=c_m$ ...
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53 views

Max of $3$-Variable Function

I'm trying the find the maximum of the function $$f(a,b,c)=\frac{a+b+c-\sqrt{a^2+b^2+c^2}}{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}$$ for all nonnegative real numbers $a, b, c$ with $ab + bc + ca > 0$. I ...
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66 views

Armijo conditions vs Reduction Conditions in Non-Linear Line Search

Overview Line search typically consists of four stages: Direction: Search direction Initial Step Size: length to search along the line on the first sub-iteration Bracket: find an interval along the ...
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165 views

Finding Shortest distance between a Sphere and Ellipsoid?

Suppose that ,I have a Sphere and an ellipsoid as Sphere: $(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2 = R_1^2$ Ellipsoid: $\large\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$ ...
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Solve non-linear equation with Matrices

I'm looking more for hints than specific answers, although I would be extremely grateful if provided with one. The problem I have is as follows: $$ -\Sigma (A+\Lambda_1)+I=0 $$ Here A is a constant, ...
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1answer
35 views

squaring the equality constraints

When creating an unconstrained optimization problem from an equality constrained one, the usual way to build the Lagrangian, is by adding a term consisting of a multiplier, multiplied by the equality ...
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1answer
37 views

Quadratic Problen with 2 constraints

Could someone help me to solve the following: $\min x^Tx$ s.t. $x^T a=1$ $x^T b=0$ where $x$,$a$ and $b$ are $(N\times1)$ vectors and $1$ and $0$ scalars. Thank you!
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Gradient descent via polynomial approximation

It seems that most proofs of convergence for gradient descent algorithms rely on strong conditions on the first and second derivatives of the function, for instance that $$|f''(x)| \leq K$$ over the ...
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1answer
117 views

Non-linear least squares with two dependent variables

I have data in the form $(t_i,x_i,y_i)$, i.e. position in 2D as a function of time. I have non-linear equations which I want to fit to the data. They give me a position $(X,Y)$ as a function of time ...
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95 views

Determining initial values for optimization problem

I am trying to solve an optimization problem with a quadratic objective function and non-linear constraints, using SQP (Sequential Quadratic Programming). I am attempting at doing the implementation ...
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1answer
274 views

Properties of the positive definite Hessian matrix of a convex function

I'm reading about nonlinear programming and I'm having trouble understanding the cool properties that a positive definite Hessian matrix $Q$ of $n$-dimensional function $f: \mathbf{R}^n\rightarrow ...
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Can difference of the $log$ function be approximated?

I am currently trying to optimize a problem. $$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$ Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to ...
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129 views

Eliminate 2 variables from 3 equations with lots of parameters

I want to eliminate the variables x and y from these 3 equations in a way that all parameters appear in one equation without x and y: ...
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98 views

sum of logarithms of linear-fractional functions Optimization Problem

I am new to optimization theory and I am facing this optimization problem. \begin{equation} maximize \qquad f(x) = \sum_{i} ...
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Strong duality: When does the optimal primal variable coincide with the primal variable giving the dual function.

I'm considering the inequality-constrained optimization problem of finding $$ x^{\star} = \arg \min_{x} f(x) \;\; \text{s.t.} \;\; h(x) \le 0 $$ which is assumed to have a unique minimizer. The ...