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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How to linearize this optimization problem?

I have a nonlinear optimization problem with linear constraints. How to solve this? $\sigma_i$ and $\rho_i$ are the optimization variables. ...
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30 views

Optimization problem: $\min \limits_{\mathbf{q}} \sum_{n=1}^N q_n$, s.t. $\frac{c_{nn} q_n }{\sum_{m \ne n} c_{nm} q_m } \ge a$

\begin{array}{rl} \min \limits_{\mathbf{q}} & \sum_{n=1}^N q_n \\ \mbox{s.t.} & \frac{c_{nn} q_n }{\sum_{m \ne n} c_{nm} q_m } \ge a, \forall n \in \{1,\ldots,N\} \end{array} For this ...
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How to solve an optimization problem with objective function as a time average expectation

I have an optimization problem with the objective as $$ \overline{h(x)}=\lim_{T \to \infty} \sum_{t=0}^{T-1} E[f(x)] $$ where $$ h(x)=\frac{f(x)}{g(x)} $$ and $f$ is convex and $g$ is linear. I ...
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1answer
29 views

Non-convex QCQP

Consider the following optimization problem: $$\begin{array}{ll} \text{minimize} & \mathbf{x}^{T} \mathbf{A} \mathbf{x}\\ \text{subject to } & \mathbf{x}^{T} \mathbf{P}_i \mathbf{x} > 0, ...
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1answer
72 views

Show that $(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4 \le 6$ for $a^2 + b^2 + c^2 + d^2 = 1$.

For $a, b, c, d \in \Bbb R$ such that $a^2 + b^2 + c^2 + d^2 = 1$, show that $$(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4 \le 6.$$ The answer uses the mysterious identity ...
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1answer
10 views

infinte dimensional nonlinear optimization

Is there any structured way to tackle $$\min_{\{x_n\},\{y_n\}}\frac{(\sum_{n=1}^{\infty}x_ny_n)^2}{\sum_{n=1}^{\infty}y_n^2\sum_{n=1}^{\infty}x_n^2}$$ when $x_n, y_n \neq 0$, ...
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1answer
46 views

Trying to solve $\max \limits_{\mathbf{x}} \sum_{i=1}^K \log_2(1+\frac{x_i a_{ii} }{\sum_{n \ne i} x_n a_{ni} })$, s.t. $\sum_{i=1}^K x_i \le b$

I am trying to solve the following optimization problem: \begin{array}{rl} \max \limits_{\mathbf{x}} & \sum_{i=1}^K \log_2(1+\frac{x_i a_{ii} }{\sum_{n \ne i} x_n a_{ni} }) \\ \mbox{subject to} ...
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36 views

Good text book recomendation

I would like to do some reading about a technique called sequential convex programming. There is a lot of material about sequential quadratic programming out there, including books (Nocedal & ...
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23 views

Find the “optimal” placement of $n$ points in a polygon

I was hoping someone could help me with a question I've been pondering the past few days. I searched the usual places (google, journals, texts, etc) but couldn't find anything that fit the bill, but ...
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3answers
99 views

max of $e$ with $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$ [closed]

Given that a,b,c,d,e are real number such that: $\begin{cases} a+b+c+d+e=8\\ a^2+b^2+c^2+d^2+e^2=16 \end{cases}$ determine the maximun value of $e$. I started like that : ...
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0answers
13 views

Indefinite Boolean Quadratic Programming: number of minima

The Boolean Quadratic Programming problem is defined as: $\min_{x} f(x) = x^TQx + c^Tx$ s.t. $ x \in \{0,1\}^n$ It is a well-studied NP-Hard problem with many approximation algorithms proposed. I ...
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6 views

Convexification of a specific constraint [closed]

Is there a way to convexify the equations of the form $y f(\mathbf{x})\leq c$ where $f(\cdot)$ is a convex function of vector $\mathbf{x}$ and c is a constant? Does it work if we further restrict ...
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18 views

Solving optimization problem where objective function is of type (affine+(affine/affine))

I need to solve a non linear optimization of the form minimize $f(x) +\frac{g(x)}{h(x)}$ subject to $p(x)\leq0$ $q(x)=0$ Here $f,g,h,p,q$ are affine functions of $x$ and they are convex in the ...
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0answers
42 views

Knapsack or bin packing problem?

I have $i$ items and I should pre-packed $m$ knapsacks with identical items where only $K<n$ items can be packed. Also, we should have only one of each item in each sack. The time capacity for ...
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0answers
28 views

Is this Feasibility problem NP-hard?

I am trying to solve a combinatorial optimization problem (a feasibility problem) and I have very little idea of solving such problems. The problem is as follows: Solve for $\phi$; \begin{equation} ...
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0answers
11 views

Estimate parameters in min function $U(a,b)$ = $min[(b-x_i)^2,(a-x_i)^2]$

I have got a list os measures and I need to estimate 2 sizes for this measures. I have the following function that estimates this 2 sizes. $U(a,b)$ = $min[(b-x_i)^2,(a-x_i)^2]$ being $a$ the first ...
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1answer
24 views

Newton's method for unconstrained minimization

Let $f(x) = \frac{1}{2} x^T Q x + b^T x + c.$ Prove that Newton's method finds a critical point after a single iteration. Here $Q$ is positive definite. For this: I need to find first of $\nabla ...
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24 views

How to solve exponential matrix factorization with constrain: $UV^T>0$

recently I would like to optimize the following loss function: $$L=\sum_{ij}W_{ij}(X_{ij}-exp(-\sum_{l} U_{il}V_{jl}))^2$$ $$s.t. \sum_lU_{il}V_{jl} > 0$$ Where $W \in \mathbb{R}^{m \times n}, X ...
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1answer
9 views

Creating gradient functions based on model parameters?

I am using a software library (Math.Net) to try to fit two Lorentzians to a curve. I have found some example software which shows the fitting out a few various types of curves (Line, Parabola, Power ...
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2answers
34 views

Optimization of a function over probability distributions

I'm trying to solve certain optimization problems dealing with probability distributions. Consider the space of probability distributions $\{ 1, ..., N\} \to [0, 1]$ I have a function $f : (\{ 1, ...
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0answers
16 views

How to numerically evaluate a integral whose limits are functions of x (using Gauss quadrature rule)?

I am trying to numerically evaluate an integral $\int_q^1 \ln (\sum_i \alpha_ix_i) dq$, in which $\ln (\sum_i \alpha_i x_i)$ is related to $q$ via the following: $z_i=(1-q)\frac{\alpha_ix_i}{\ln ...
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0answers
18 views

Minimizing convex functions without compatible gradients

I've been working on a minimization problem for a while, involving "simple" conditions, but haven't been able to figure it out. I've tried using Lagrange Multipliers and KKT, but the presence of ...
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0answers
31 views

An optimisation problem

I have an optimisation probem given below $$argmax_{x_i \ \ \forall x_i=1,2...n} \sum_{i} S_ie^{-\alpha x_i}$$ subject to $$\sum x_i = 1$$ $$\sum C_i x_i \leq B $$ $$\forall i \ \ x_i \geq 0 $$ ...
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0answers
20 views

Nonlinear equations algorithm - Newton method

Some time ago I posted a question regarding the simple case of finding the intersection point when I have only two functions, and with your help I found an answer. It was this case: $f(x) = a + ...
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2answers
69 views

Finding the maximum area of a triangle with a perimeter constrain

Using graphical methods, determine the dimensions of a right triangle that has the largest possible area, given that the perimeter cannot be larger than $P$. The final answer should be in terms of ...
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1answer
26 views

How to find matrix $A$ from the relation: $A\times (A^TA)^{-1}\times A^T = B$

Kindly help me in the following: I have two Matrices, $A$ of size $(n\times m)$; and $B$ of size $(n\times n)$, where $n>m$. $A$ is unknown, but $B$ is known. $(A^TA)$ is invertible $B$ is ...
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0answers
42 views

Solving optimization involving square roots?

I am new to optimization and I have the following problem that I would like to analyze and obtain a good solution (if optimal solution cannot be reached) $$ \max_\mathbf{x} \quad \sum_{n = ...
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1answer
25 views

show that $\Big(\sum_{i=1}^{n}\alpha_i ^2\Big)^2\leq\Big(\sum_{i=1}^{n}\alpha_i \Big)\Big(\sum_{i=1}^{n}\alpha_i ^3\Big)$

Let $\alpha_1,\alpha_2,...,\alpha_n>0.$ How can I show that $$\Big(\sum_{i=1}^{n}\alpha_i ^2\Big)^2\leq\Big(\sum_{i=1}^{n}\alpha_i \Big)\Big(\sum_{i=1}^{n}\alpha_i ^3\Big).$$ Please provide me ...
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15 views

Approximating a quadratic term in the constraint set as 2nd order Taylor expansion

I have an optimization problem in the following form: $$\min_{x,y} f(x)+g(y)$$ $$s.t.$$ $$Ax+h(y)=0$$ where $h(y)$ is a quadratic in $y$. Instead of solving this problem directly, $h(y)$ is ...
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1answer
21 views

What is the class of this Integer programming prob.

I have an optimization problem which seems to be non-linear because of the constraints (right?): $max (\sum U_i\times x_i)\\ \sum x_i\times y_i\times r_i\leq R\\ \sum y_i=1\\ \sum x_i=1\\ x_i, ...
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20 views

Coordinate descent with equality and inequality constraints

I have an intuitive understanding of why the simple method of coordinate descent does not work with linearly coupled constraints such as; $$\min_x\sum_if_i(x_i)$$ $$s.t.$$ $$Ax=b$$ If we try to ...
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25 views

Optimization problem regarding Newton's algorithm

I would want to ask why does Newton's algorithm with Wolfe line search converges to (0,0) no matter where the starting point is?
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1answer
53 views

Max $z = x_1(1-x_2)x_3$ s.t. $x_1 - x_2 + x_3 \le 1$

Using dynamic programming, Maximise $$z = x_1(1-x_2)x_3$$ subject to $$x_1 - x_2 + x_3 \le 1$$ $$x_1, x_2, x_3 \ge 0$$ Here's the outline of my solution 1. How is it? Let ...
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6 views

Minimizing the distance between two set of vectors such that the angle of both set is equal

Suppose I have two set of vectors K1,I1 and K2,I2 forming a surface S1 and S2 respectively in R2 or R3. The angle between K1 and I1 is T1 and K2 and I2 is T2 respectively. The goal is to minimize the ...
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0answers
10 views

Nonlinear regression output in R

Suppose one is interested in doing a nonlinear curve fitting procedure such as $Y=AX^B$ where $a,b \in \mathbb{R}$. If the regression were linear, one usually observes the standard error of the ...
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1answer
21 views

Optimisation Problem [closed]

Given a vector $\vec{c}$ and a radius $r$, solve the problem: \begin{equation*} \begin{aligned} & \underset{x}{\text{maximise}} & & \vec{c} \cdotp \vec{x}=a \\ & \text{subject to} ...
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1answer
20 views

$L^p$-norm minimization under linear constraints: Does the optimum depend on $p$?

Consider the following norm minimization program: \begin{align} \label{1} &\min_{x \in \mathbb{R}^d} &&\lVert x - x_0 \rVert_p^p &(1)\\ &\text{subject to } &&Ax-b \ge 0 ...
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23 views

Rosenbrock function matlab

I am new to MATLAB and I am asked to implement on matlab the following algorithm: for an unconstrained minimisation problem. I am asked to apply the BFGS method with armijo line search ...
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0answers
50 views

An interesting optimization problem with a quadratic equality constraint

I am trying to find a closed-form solution to an interesting optimization problem, which seems to be simple, but not trivial in fact. OK, here is the problem: min$_s$ Re($v^Hs$) s.t. ...
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31 views

Residual norm of active set method in non negative least squares

I am having trouble with understanding one of the statements in active set methods done for non negative least squares given in Book written by Lawson. The quadratic problem can be written as ...
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2answers
272 views

Optimal path around an invisible wall [duplicate]

The Problem On an infinite plane there are two points, $A$ and $B$, a unit distance apart. There is a $50\%$ probability that there is an invisible wall somewhere between the two points. The ...
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0answers
15 views

Solving non-linear functional equations numerically by sequence of linear least-squares?

So I am experimenting with a linear systems solver to find new exciting applications for it. While it is possible to play around to solve some of the more basic functional equations, I am trying to be ...
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28 views

Gradient descent method for real function of complex matrix

Suppose $\mathrm{a}$ is $N\times 1$ known complex vector, and we need to solve this following optimization problem with the gradient descent method: ...
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1answer
10 views

How to optimize a non-convex function with nonlinear equality constraints

\begin{equation} \min \|X-UV\| \end{equation} \begin{equation} \textbf{s.t.} \|Uz_i\|_{2}=1 \end{equation} with $V_{n\times m}=(z_1,z_2,...,z_m), z_i=(v_{i1},v_{i2},...,v_{in})^{\top}$
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26 views

Drift management optimization

I have a problem in which I am having trouble formulating the optimization. A portfolio value is $10M I have a vector of current weights [.10,.15,.15,.10,.05,.10,.20,.15] and another vector of ...
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12 views

Exact and Heuristic Optimization Methods

Could anyone give me a rough classification for which kind of nonlinear- problems can I apply exact optimization methods (such as barrier function) and for which problems heuristic methods (such as ...
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0answers
7 views

What exactly are convex constraints?

I haven't been able to find a clear answer to this question, seek an answer from a professor or figure it out myself as I am not a mathematics expert. I used the ...
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0answers
7 views

Non-Convex fractional problem of maximizing rate over power

Dears, I have a non convex fractional maximization problem. I have the optimization variable both in the numerator and denominator. It's a problem of maximizing the sum rate of a network divided by ...
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0answers
48 views

Derivative error for Lagrange interpolation

I was reading a book and I found this (with some context): if $f(x)=L(x)+R(x)$, with $L$ the quadratic interpolation with three points $x_0, x_1$ and $x_2$, then $R(x)=\dfrac{f'''(\epsilon(x))}{6} ...
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1answer
24 views

Minimization problem with infinite variables and linear constraints

How can this minimization problem be solved? $$ \left\{\begin{matrix} \begin {aligned} &\sum_{i=1}^{\infty}P_i^3 \rightarrow min \\&\sum_{i=1}^{\infty}P_i=1 \\ &P_i\geqslant 0 \:for\: ...