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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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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48 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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14 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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17 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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1answer
24 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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37 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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2answers
287 views

Optimization of Frobenius Norm and Nuclear Norm

How to solve the following optimization problem, \begin{equation} \boldsymbol{\hat{x}} = argmin_{\boldsymbol{x}} \frac{1}{2} \| \boldsymbol{x - y} \|_F^2 + \lambda \| \boldsymbol{x} \|_{*} ...
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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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1answer
20 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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18 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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1answer
46 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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9 views

Sequential convex second order cone programming [on hold]

I am trying to solve an optimization problem where the objective function is convex, the inequalities are second order cones but I also have nonlinear equality constraints. The convex energy can be ...
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0answers
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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2answers
152 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
12 views

How to call the function from CUTE library in MAtlab interface? [closed]

There are CUTEr/CUTEst Library for the test functions of unconstrained and constrained optimization problem. I need to call those functions in symbolic form in Matlab interface. Please help me in ...
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1answer
465 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) ...
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number of constraints is a variable [closed]

I have a mixed integer programming problem. Some of my variables take fixed value (say 0), which is dependent on the integer variable. Are there any references to this kind of problem. The objective ...
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4 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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9 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
20 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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28 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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1answer
17 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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13 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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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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0answers
26 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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24 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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2answers
215 views

Optimization of Product of Different Objective functions (Ex.: Maximize The Product of projections of a complex vector)

Suppose We have this optimization problem which is convex $\mathbf{x}={\arg}\: \underset{\mathbf{x}}\max f_{i}\left (\mathbf{x} \right )$ But the product of different objective function is ...
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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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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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6 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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1answer
102 views

Solution for $\min_{x^Tx=1} x^TAx-c^Tx$

$\min_{x^Tx=1} x^TAx-c^Tx$ looks like a simple QPQC problem. If $A$ is positive semi-definite, can I get the solution by first getting $x=A^{-1}c$ and then projecting $x:=\dfrac{x}{||x||}$ to make ...
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40 views

Find a solution of optimal problem with an inequality constraint

Let $a,b,x$ be vectors in $R^n$, A be a matrix, $c,d \in R, c<d$. Solve the following problem: $$\begin{cases} \text{minimize} \quad (b-Ax)^T(b-Ax)\\ (a^Tx-c).(a^Tx-d) \leq 0 \end{cases}$$ Assume ...
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46 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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43 views

Nonlinear constraints replaced by parameters and estimated iteratively

I have an optimization problem with nonlinear constraints in the following form: $x + y + 0.5(x+y)^2-z = 0$ $s+(x+y)*t\ge M$ I linearize these constraint by replacing the nonlinear terms by ...
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12 views

Which are the alternative approaches to stochastic (online) gradient descend for online optimization?

I'm looking for some alternative approaches to online\stochastic gradient descent for online optimization such that 1) there exists some proof about the convergence of the parameters to some compact ...
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1answer
32 views

Linearize non-linear constraint [closed]

I have a problem which may be defined as: $$\max 5 x_{11} + 6 x_{12} + 2 x_{21} + 3 x_{22} \\ x_{ij}\in \{0,1\} \\ x_{11} + x_{12} = 1 \\ x_{21} + x_{22} = 1 \\ t_1,t_2 \text { integer} \\ (t_1 - ...
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1answer
20 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\: ...
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385 views

Local optimality of a KKT point.

Consider the problem \begin{equation} \min_x f(x)~~~{\rm s.t.}~~~ g_i(x)\leq 0,~~i=1,\dots,I, \end{equation} where $x$ is the optimization parameter vector, $f(x)$ is the objective function and ...
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18 views

Can the constrained optimization problem (1) be transformed into the unconstrained form (2)

(1) \begin{equation}\label{constrained} \begin{array}{cl} \arg \min \limits_{\mathcal{C}_k} & \text{rank}(\mathcal{C}_k)\\ \mathrm{s.t.} & \mathcal{E}(\phi_{j}^{k})\le \epsilon ...
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24 views

Are the constrained optimization problem equal to the unconstrained one?

(1) \begin{equation}\label{constrained} \begin{array}{cl} \arg \min \limits_{x} & \|Ax-b\|_2\\ \mathrm{s.t.} & \|x\|_1\le \epsilon \end{array} \end{equation} (2) ...
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Why the trace norm of a tensor is a good approximation to its rank?

In many literature, they use the trace norm of a tensor to approximate its rank, which is defined as follows: $$\|\mathcal{X}\|_{\ast}:=\sum_{i=1}^{n}\alpha_i\|X_{(i)}\|_{\ast}$$ where ${\alpha_i}'s$ ...
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9 views

Direction of Gauss-Newton is always descent

Gauss-Newton algorithm How would I go about proving this? For the problem $$ min \ f(x) = \frac{1}{2} \Sigma_{j=1}^m r_j(x)^2 $$ The equations for the search direction $$ J_k^TJ_kp_k=-J_k^Tr_k $$ ...
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1answer
21 views

how to find extreme points for 3 variable linear programming

It is rather easy to find extreme points in 2 variable case. But to find them for higher dimensions, for example in 3 variable case. For instance, min $-3x_1-2x_2-x_3$ st. $2x_1+x_2-x_3\le2$ ...
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1answer
32 views

Show non-convexity of a function with vector input

How does one go about proving non-convexity of the function d? $$ d(v) = 1/2*||F(v)- p||^2 $$ $$ F(v)=\sum_{i=1}^n l_i*\begin{pmatrix} cos(\sum_{j=1}^i v_j) \\ sin(\sum_{j=1}^i v_j \end{pmatrix} $$ ...
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8 views

Maximize nonlinear nonconvex optimization

Nonlinear nonconvex maximization problem Signal to noise plus interference ratio Please how do I resolve a problem of this nature?
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11 views

How to find the Lagrangian dual function (for three variables)?

How to find the Lagrangian dual function: min$-3x_1-2x_2-x_3$ s.t. $2x_1+x_2-x_3-2\le0$ $x_1+2x_2-4\le0$ $x_3-3\le0$ $x_1,x_2,x_2\ge0$ over $X=\lbrace ...
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12 views

Nonlinear Optimization SQP method

I have a question about non linear optimization and the SQP method. Assignment a) Derive the KKT system $\nabla_xL(x,\mu) = (-1,-1)+2\mu x = 0$ and from the equality constraint we have $\Vert ...
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17 views

How to find extreme directions?

objective:min $−3x_1−2x_2−x_3$ The set is : $X=\lbrace (x_1,x_2,x_3):2x_1+x_2-x_3\le2; x_1,x_2,x_3\ge0 \rbrace$ Attempt: $2d_1+d_2-d_3\le0$ (a) $d_1+d_2+d_3=1$ and $d_1,d_2,d_3\ge0$ Since from ...
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10 views

separability of dynamic programming

I am working on some portfolio selection problem and running into this concept. It is stated that "multiperiod mean–variance formulations cannot be solved using dynamic programming due to their ...