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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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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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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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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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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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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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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28 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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58 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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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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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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51 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
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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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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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19 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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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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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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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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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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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 f(x^{(k)...
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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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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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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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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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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 \{0, ...
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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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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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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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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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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 \end{bmatrix}^{...
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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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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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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} \displaystyle\sum_{i=1}^n\ln\left(1+a_i\...
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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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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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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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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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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 $b_i \in (0,\infty)$ ...