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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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 $y_2=...
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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
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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
25 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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47 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 (backtracking)....
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53 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. $||s||_2=...
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34 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 \begin{...
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2answers
342 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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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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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: $\mathrm{X}=\underset{\mathrm{X}}{\mathrm{arg\,min}...
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12 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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27 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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15 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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9 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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49 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} (x-...
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27 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\: i=1,...
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43 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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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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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 \end{...
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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) \begin{equation}\label{...
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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
36 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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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 x\...
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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 (x_1,x_2,x_2):2x_1+x_2-x_3-2\le0;x_1,x_2,...
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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. $2x_1+x_2-x_3\le2$ $x_1,...
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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 ...
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33 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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1answer
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Trust-region method

The question has to do with the trust-region method for unconstrained optimization. I came across it on p.~392 of Linear and Nonlinear Optimization, by Griva, Nash and Sofer. Let $p(\lambda)$ be ...
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1answer
109 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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What is the initial tableau for simplex method with big M method for this problem?

I have an optimization problem with formulation: min f = x1+x2+x3 subject to: x1+2*x2+x3=8 2*x1+x2+x3=12 x1,x2,x3>=0 I should solve it by Big M method. For this I added two extra variables (a1,...
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Scaling vector-valued objective function for non-linear optimization/minimization

I am trying to minimize a non-linear vector-valued function in MATLAB. As a test case for my code, I try to minimize a function whose solution I know apriori. The problem is that one of the solutions ...
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41 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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1answer
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What kinds of optimization it is? (with indicator)

I don't know what kinds of programming model it is with an indicator function in the constrained. Thanks for providing any keywords! Maxmize $30R_1+20R_2+12R_3+15R_4$ Subject to: $0\leq R_{1} \leq ...
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1answer
53 views

Adding constraints in a constrained problem

Consider a simplified version of a problem I am looking at: $$\min_{x, y, z, t_1, t_2, t_3} x - x^2 - y + y^2 - z + z^2 + t_1$$ subject to: $$ -x + x^2 \leq a + t_1$$ $$ -y + y^2 \leq b - t_2$$ $$ -z +...
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Previous research on nested optimization: sensor allocation and sensor placement?

I'm working on a project in my optimization class and have come across the Weapon Target Assignment Problem (WTA) and the Art Gallery Problem (AG) I want to apply WTA to the problem of optimally ...
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How can I solve this as an optimization problem?

I would like to find x such that (Ax).^2 + (Bx).^2 == I (using Matlab syntax). A, B are matrices and I is a vector, all with real values. The number of equations is less than the number of variables, ...
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Non-Linear Optimization

Hello I'm trying to do a non-linear optimization assignment, which causes me some trouble Assignment Answers so far: a) $X = \lbrace x \in R^n : g(x) = 1-\|x\|_2^2 \leq 0 \rbrace$ is a concave ...
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1answer
30 views

Solving nonconvex problem by iterating convex ones

I have a convex problem with the following properties: -The energy to be minimized is convex - it is basically a norm. -The domain is defined by a set of convex cone constraints inequalities. I am ...
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70 views

Non-Linear Optimization calculate Critical cone

I have a problem with understanding the critical cone in a nonlinear optimization assignment: $f(x) = -(x_1-1)^2-(x_2+1)^2, g(x) = (-x_1,x_2,-x_1x_2)^T, \hat{x} = (0,0)^T\\ \mathrm{min}_{x \in \...
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15 views

Maximum Piecewise-Linear Lower-Bound

I've run into a problem, which I've formalized here: Given: $n \geq 1, m \geq 1$ fixed integers. $x_0, \cdots, x_n$ evenly spaced list of reals, with $x_0=0, x_n=1$. $y_0, \cdots, y_n$ ...
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Minimizing percentiles of discrete distribution

I have a vector $\vec{v} \in \mathbb{R}^n$ and a matrix $A \in \mathbb{R}^{n \times m}$. For any $\vec{x} \in \mathbb{R}^m$, the vector $\vec{v} + A\vec{x} \in \mathbb{R}^n$ represents a discrete ...
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circular cones of real-valued random variables

My professor gave me a problem that if we consider the vector space of all real-valued random variables can we define a norm where with it we can construct a circular cone which is symmetric? If we ...
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18 views

How to calculate the Bouligand derivative (B-derivative)

Let $H(x)=\min (f(x),h(x))$ where $f$ and $h$ are continuously differentiable functions from $\mathbf{R}^n$ to $\mathbf{R}^1$. The Bouligand derivative (B-derivative) $BH(z)$ at $z$ of $H$ is given ...
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Newton's Method stalls upon reaching feasible point

I'm implementing Newton's Method to solve a nonlinear program with general equality constraints, i.e. \begin{equation} \underset{\mathbf{x}}{max} \ f(\mathbf{x}) \end{equation} \begin{equation*} s.t ...
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63 views

Bilinear (non-convex) objective and linearized (big-M) constraints

My original problem is a MIQCQP. The bilinear terms in the constraints are products of binary and continuous variables and can be linearized using big-M. The bilinear terms in the objective function ...
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2answers
25 views

Intuitively, why does squaring a loss function change optimal values?

In many optimization problems, it is clear that by performing a non-linear operation we change the outcome of any potential optimal values. For example in machine learning: summing over errors (...
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Optimization with an non-linear ODE constraint

I want to optimize (minimize) such a cost function: $$\min_{x_1, x_2, \ldots, x_t} \left(\sum^t_{k=1} (y_k - \theta_k)\right)$$ where $\theta_k$ is some pre-defined constant and $dy/dt = f(x)$. You ...