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0
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0answers
19 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
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0answers
14 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
26 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
20 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
0answers
46 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
45 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
30 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 ...
-1
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0answers
34 views

Nonlinear equations with boolean variables

Let $i = 1, \dots, v$, $j = 1, \dots, v$ and $n = 1, \dots, N$. $i$ and $j$ indicate origin and destination nodes in a graph, respectively. An individual is denoted by $n$. Also let $0 < \alpha, ...
2
votes
0answers
78 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
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0answers
28 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
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0answers
19 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 ...
0
votes
1answer
30 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 ...
3
votes
2answers
156 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 ...
0
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0answers
15 views

A non-linear optimization problem involving logarithms

I have a log likelihood function that I'm trying to minimize: $-\log(L) = -(\sum_i^n{c_i \log(a_i x + b_i y)} - \sum_i^n{\log(c_i!)} - x \sum_i^n{a_i} - y \sum_i^n{b_i})$ The function is only ...
1
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1answer
43 views

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 ...
0
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1answer
29 views

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 ...
-1
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0answers
36 views

Ellipsoid-Sphere Intersection [duplicate]

Suppose I have an Ellipsoid given as: $$\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$$ And a Sphere $$(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2 = R_1^2$$ If they intersect ...
0
votes
1answer
52 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 $s_1 : g_1(x_1,x_2,...,x_n)=c_1$ ... $s_n : g_m(x_1,x_2,...,x_n)=c_m$ ...
3
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2answers
51 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 ...
0
votes
1answer
12 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 ...
0
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1answer
62 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: $\large\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$ ...
3
votes
0answers
27 views

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, ...
0
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1answer
19 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 ...
1
vote
1answer
28 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!
1
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0answers
27 views

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 ...
1
vote
1answer
71 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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0answers
31 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 ...
-1
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0answers
28 views

how to calculate least square root of a nonlinear equation?

I have $z= x*y$, and I want to calculate the least square root of this equation. So it becomes: $(z+dz)^2=(x+dx)(y+dy)$ then I can continue with replacing $z$ with $x\cdot y$ and $dz$ with $dxdy$ but ...
1
vote
1answer
45 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 ...
1
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0answers
26 views

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 ...
2
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1answer
50 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: ...
0
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1answer
40 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} ...
0
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0answers
15 views

First and second derivatives of barrier term in a quadratic programming problem

I am implementing an algorithm of Dang and Xu's, ``Non-convex Quadratic Programming Problem with Box Constraints'' and I'm hoping that somebody could verify what I'm doing. Their algorithm minimizes ...
3
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0answers
38 views

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 ...
1
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1answer
26 views

Do we need steepest descent methods, when minimizing quadratic functions?

I'm studying about nonlinear programming and steepest descent methods for quadratic multivariable functions. I have a question highlighted in the following picture: My question is: If we can ...
3
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2answers
39 views

The quadratic case in nonlinear programming

I'm reading about nonlinear programming and I stumbled into the following statement where I started to wonder a bit: Consider the function $$f(\textbf{x}) = ...
0
votes
1answer
20 views

Goldstein test in nonlinear programming

I'm reading about nonlinear programming and the Goldstein test. Here is the definition from my book: A line search accuracy test that is frequently used is the Goldstein test. A value of ...
0
votes
1answer
26 views

Above what order of magnitude a pure cutting-plane algorithm must be forgotten in favour of branch-and-cut?

Crawling the web on the subject of the cutting-plane algorithm, I have seen everywhere that a pure cutting-plane method cannot be used for numerical instability reasons after some iterations. But do ...
1
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1answer
28 views

Can Gomory's cutting plane be used to solve Mixed Integer Linear Programs?

Do you know if Gomory's cut can be used to solve MILP problems ? I have read that Gomory's cut is useful when all variables are integer, but what if just some of them are integer ? Is is necessary ...
1
vote
1answer
52 views

Matrix Maximization

I would like to solve the following optimization problem for a matrix $X$ which is symmetric and positive-semidefinite: $$ \mathrm{maximize} \, \, \, f(X) = \log \mathrm{det} X - k_1 \log(k_2 + a^T X ...
1
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1answer
33 views

Order of convergence for the method of false position

I'm reading about the order of convergence of the method of false position and there is one tricky point in the proof I don't understand. The method itself for finding the minimum $x^*$ of a function ...
1
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1answer
25 views

Maximizing the probability of multivariate hypergeometric distribution

Suppose there are $R$ number of red balls, $B$ number of black balls, and $W$ number of white balls in an urn. The total number of balls are $N$, thus, $N = R + B + W$. I draw $M$ number of balls from ...
1
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0answers
28 views

Zero-order necessary conditions

I have a question regarding the Zero-order necessary conditions. In my Linear and Nonlinear programming book it is stated: Consider the set $\Gamma \subset E^{n+1} = \{(r,\textbf{x}): r\geq ...
0
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0answers
20 views

Minimization problem with amplitude constraint

I have the following minimization problem: $$\left\| \bf{A}x - y\right\|^2 \to min $$ $$s.t. \left|x_i\right| < 1, \forall i,$$ where $\bf{A}$ is the complex matrix with size of $(n\times m)$, ...
0
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1answer
22 views

Optimality conditions in convex programming

I'm reading about Zero-order conditions in Nonlinear Programming and the following confuses me (my questions are below the theory): Consider the set $\Gamma \subset E^{n+1} = \{(r,\textbf{x}): ...
1
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0answers
39 views

implicit non-linear equations with complex variables

I am trying to understand a methodology for solving implicit non-linear equations with complex variables. I would like to solve for z1 below where z2 is known. Also both z1 and z2 are complex ...
0
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2answers
52 views

Nonlinear Optimization problem

Function $f(x) \in \mathbb{R}^n$, $(n\geq 1)$, depend on one parameter $x \in \mathbb{R}$. Performing a nonlinear transformation of $f(x)$, we obtain function $g(y) \in \mathbb{R}^n$. This ...
0
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2answers
60 views

Optimizing elementary symmetric polynomial on the unit sphere

I'd like to optimize $x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4$ on the unit 4-sphere. I'm thinking I should do lagrangian optimization, but I'm having trouble solving the resulting ...
0
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2answers
33 views

optimization problem: finding an hyperplane separating one point from a set of pointy maximizing the distance

I have this problem: I have a set of n-dimensional points $P$. I have one more n-dimensional point $q$. The points in $P$ are linearly separable from $q$ (i.e. it always exists an hyperplane $n^t x ...
1
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2answers
69 views

Forbidden range for a linear programming variable

I would like to express a linear program having a variable that can only be greater or equal than a constant $c$ or equal to $0$. The range $]0; c[$ being unallowed. Do you know a way to express this ...