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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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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26 views

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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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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1answer
41 views

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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11 views

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
25 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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31 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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13 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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9 views

Non Linear Optimization Abadie constraint qualitfication

I have a non linear optimization problem: In (b) I find that $$T(X,\bar{x}) = \lbrace x \in R^2 : x_1 \geq 1 , x_2 \in R\rbrace$$ and $$T_l(g,\bar{x}) = \lbrace x \in R^2 : x_1 \geq 0 , x_2 \in R ...
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5 views

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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13 views

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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15 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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44 views

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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25 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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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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13 views

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 ...
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18 views

Binding inequality constraints in linear programming with quadratic constraints

I am trying to maximize the following objective function: $a_{1}b_{1}x_{1}+a_{2}b_{2}x_{2}+a_{3}b_{3}x_{3}+a_{4}b_{4}x_{4}$ The quadratic constraint is given by $b_{1}^2 x_{1}^2 + b_{2}^2 x_{2}^2 + ...
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64 views

Nature of the Hessian of the dual function?

I originally posted this over at MathOverflow but it did not receive much (...any) attention. I'm hoping someone can point me in the right direction over here. Consider a nonlinear optimization ...
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1answer
27 views

Nonlinear integer programming problem

I am trying to maximize the following function $$ f(m,n) = \frac{m \log 3 + n \log 2}{\sqrt{m^2+n^2}} $$ where $ n $ and $ m $ are integers, not both $ = 0 $, although one could be $ 0 $. This is ...
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20 views

trust region - choice of scaling matrix

According to many resources, TR algorithms often suffer from bad scaling. The simplest remedy is to use scaling matrix D in following way \begin{align} \min_d \ f + g'd + \frac{1}{2}*d'Bd \\ ...
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15 views

Can the low-rank approximation problem be formulated as the following convex model?

Given a three-order tensor $\mathcal{Y}$, our aim is to find a tensor $\mathcal{X}$ to approximate it and $\mathcal{X}$ should satisfy the following property: $\mathcal{X}$ can be well approximated ...
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7 views

Nonlinear Programming Problem and Projected Gradient Algorithm

Consider $\max f(x)$ subject to $||x||=1$ where $f(x)=\frac{1}{2} x^TQx$ and $Q=Q^T$. We want to apply a fixed step gradient algorithm to this problem: $$ x^{(k+1)}=\Pi(x^{(k)} + \alpha \nabla ...
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37 views

Quasiconvexity of a function on the positive orthant using rays

It is seen in Boyd's book on Convex Optimization book that to show a function $f:\mathbf{R}^n\rightarrow \mathbf{R}$ is quasiconvex, it is enough to show that $f$'s restriction to a line is ...
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27 views

Proving equivalent optimization problems

Consider the problems $\min f(x) , x \in X$ and $\min g(x), x \in X$. two optimization problems are said to be equivalent if an optimal solution to one, is also optimal to another. I would like to ...
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1answer
28 views

Convexity of n $4u^4x^{2.5}y^{-5}$ over $x>0, y >0$ and $u>0$

I am trying to find if the function $4u^4x^{2.5}y^{-5}$ is convex over $u>0, y>0$ and $x>0$. The thing which comes to my mind immediately is to check the positive semi-definiteness of the ...
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13 views

School District Boundary Optimization--Interpreting the Objective Function

I’m looking for a little help on a new problem. I’m in a linear programming class and trying to work on a project exploring methods on nonlinear optimization and I came across the following question ...
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1answer
35 views

Convert non-linear into linear

What I tried: Let $$u_1 = x_1^3$$ $$u_2 = x_2 x_3$$ $$u_3 = x_3^3$$ Then we have $z = u_1 + u_2 + u _3$ s.t. $1 \le u_3 \le 343$ $u_3^{1/3}$ should be integer $u_1 \in \{0, 1\}$ $u_2 \in ...
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1answer
21 views

Convexity of the Pareto front: formal definition

Does anyone have a reference to a formal definition of what convexity of a Pareto front in multiobjective optimisation means? All literature I've seen uses the term without defining it.
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17 views

Convergence rate for such modified method of steepest descent

We consider only the quadratic case. $f(x,y)=\frac{1}{2}x^TQx. $ And suppose we can choose $x_0$ to make $g_0$ in the span of its eigenvectors $e_i$, where $g_k=Qx_k$, being the gradient in each ...
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19 views

Numerical Optimizer Matlab Calibration

I am trying to mimimize the following function in order to calibrate the Libor Market Model $$\sum_{i=1}^{n} \left(\sigma_i^{market}-\sigma_i^{Reb}\left(a,b,c,d,\beta\right)/\sqrt{T_i}\right)^2,$$ ...
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18 views

Minimum bounding rectangle is aligned with the convex hull

To start off, here's the problem I'm trying to solve: Suppose we have a finite collection of points in 2D. We would like to find the minimal bounding rectangle (MBR) for these points. By definition, ...
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35 views

Matrix optimization over a quadratic function

I want to find matrices $F$, $G$, and $H$ minimizing $\begin{bmatrix} x^T & y^T& z^T \end{bmatrix} \begin{bmatrix} I & 0& 0 \\ 0 & F &0 \\ 0 & G &H ...
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1answer
26 views

Lagrange multiplier - maximum not on tangent contour

I am trying to validate how Lagrange multipliers work. Looking to maximize $f(x,y)=1-x^2$ along curve $x^2 + y^2 = 1$, the solutions are $f(0, -1) = f(0,1)=1$. However, according to Lagrange ...
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Maximizing $y^H ( I - X pinv(X) ) y $ with respect to matrix $X$. How hard can it get?

Assume $X$ to be a tall block-diagonal matrix where each block is a collumn vector. Assuming $X^+ = (X^H X)^{-1}X^H $ to be the pseudoinverse of the matrix $X$, find $X$ which maximizes $$y^H ( X X^+ ...
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15 views

NLP, Recatungular or Polar? Sine or No Sine?

Let's say we have the following complex number, $V$: $V=V_i+jV_j=V_m\angle{V_a}$ Which type of representation is better in an NLP (polar or rectangular)? The polar form leads to one nasty ...
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28 views

Can we solve this system of inequalities analytically?

Let $A$ be positive real number and $k$ a positive integer. How to find the analytical solution of this system? Find the $a_i$ \begin{align} \begin{cases} ...
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1answer
59 views

Why does $f(x)=\frac{x^T Ax}{x^T x}$ always have a minimum value?

$f$ is defined for all $x\in\mathbb{R^n}-\{0\}$ nd $A$ is a symmetric matrix $n \times n$. I have to proof that $f$ has a minimum $f(x^*)$ and write a formula for $x^*$ using the spectral ...
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30 views

A supremum problem

Let $a=\underset{\|u\|_c\leq 1, \|v\|_r\leq 1}{\sup}v^TY^Tu$. If $\lambda<a$, $\underset{u_m, v_m}{\sup}v_m^TY^Tu_m-\lambda\|u_m\|_c\|v_m\|_r=+\infty$. While if $\lambda > a$, then ...
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2answers
43 views

Intersection of a power function with a line: how to compute?

How to compute $x$ from $$q x^p = 1 - x$$ where $x$ and $q$ are positive, while $p$ is a real number? When $p > 0$: it's two monotonic functions, one increasing and one decreasing, and having ...
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133 views

Nonlinear optimization: Optimizing a matrix to make its square is close to a given matrix.

I'm trying to solve a minimization problem whose purpose is to optimize a matrix whose square is close to another given matrix. But I can't find an effective tool to solve it. Here is my problem: ...
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18 views

How to regress certain non-linear data

How can I perform a regression onto data of that follows this shape: \begin{equation} U(x):=\sum_{i=1}^N\, a_ix^ie^{-b_ix} \end{equation} where the $a_i\in \mathbb{R}$ and the $b_i \in (0,\infty)$ ...
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1answer
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Proving inequality from convexity of function

I am having trouble proving the following inequality for all $x,y>0$ from "The Mathematics of Nonlinear Programming" by Pressini, Uhl. The book states that it follows from the convexity of an ...
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12 views

Condition of two related matrices

I have a data matrix $\text{X} \in R^{n \times m}$ (n - number of variables; m - number of experiements) and two parameter vectors $\beta_{p} \in R^{p \times 1}$ and $\beta_{l} \in R^{l \times 1}$. ...
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What algorithms are applicable to solve a inequality constraint Quadratic Optimization?

Suppose that we have a quadratic optimization problem $$(QP) \qquad \min \lbrace\frac{1}{2}x^TQX+ q^TX\rbrace $$ s.t. $$AX=a;$$ $$BX\le b;$$ $$X \ge 0;$$ where $Q \in \mathbb{R}^{n \times n}$ ...
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7 views

Mapping from variable space to ccriterion space in Multiobjective Linear Fractional Programming

I would like to ask about the properties of the criterion-objective space of a Multiobjective Linear Fractional Program with two linear fractional objectives for maximization and linear constraints. I ...
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1answer
22 views

Global minimum of a parameteric function

Let $q:[1, + \infty) \subset \mathbb{R} \longrightarrow \mathbb{R}$ be a function defined as $ \qquad \qquad \qquad \qquad \qquad \quad q(x) = \left \{ \begin{array}{lcl} \delta_{1} & \text{ if } ...
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mutivariable unconstrained optimization using gradient search procedure [closed]

Multi-variable unconstrained optimization problem: Maximize the function, $$f(x)=2xy+2y-x^2-2y^2$$ using the gradient search procedure.
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19 views

What is the order of convergence of a vector?

I have a vector of sequences say $(1/k, (1/k)^k)$. I know that each elements of the vector converge to 0 but the way they converge is different. the First element converges sub linearly and the second ...
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11 views

Dual of the following non-linear program

I am new to optimization and understanding some concept. I understood how duality work and tried applying it some linear programs. I followed the same for non-linear programs but I end up wth a ...