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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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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13 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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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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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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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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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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21 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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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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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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35 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 ...
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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 ...
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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. $2x_1+x_2-x_3\le2$ ...
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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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20 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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21 views

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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108 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 ...
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14 views

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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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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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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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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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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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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50 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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40 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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15 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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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 ...
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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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38 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
22 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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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 ...