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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19 views

Reducing uncertainty of a mathematical model with data (Process control)

I know this is a very broad question, but need suggestions, link to good reference papers etc. So here is the question: I have an uncertain model whose parameters are static (not changing with time) ...
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1answer
25 views

Explanation of $\textrm{argmax}_j{z_jt}$ and how to implement it

I'm struggling with one equation within a subtour elimination constraint. $$\sum_{i \in S} \sum_{j \in S, j<i} y^t_{ij} \le \sum_{i \in S} z_{it} -z_{kt} \quad S \subseteq M \quad t \in T \quad ...
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25 views

joint optimization problem with somewhat symmetric function

I have just brief question that the method that I use to solve optimization problem is legit. I have function $\max_{x,y}F(x,y)$, and first order condition gives me following equation. ...
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1answer
95 views

MATLAB: minimize function using x value from previous iteration

I'm trying to develop an algorithm for a proximal point method defined as: $$ \underset{x \in \rm I\!R^n}{\arg\min} f(x) + \lambda g(x) $$ where f(x) is a convex and coercive function and also ...
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21 views

Find $\alpha$ from equation $F(\alpha)=\int \left (\frac {I(x)}{\alpha^TG(x)}-1\right)^2 \, dx+\lambda\|\alpha\|^2$

I have a function such as $$F(\alpha)=\int \left (\frac {I(x)}{\alpha^TG(x)}-1\right)^2 \, dx + \lambda \|\alpha\|^2$$ where $I,\lambda,G$ are given. In which $G(x)$ is a vector; $G=[G_1(x), G_2(x), ...
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17 views

Help required in solving the lagrangian dual?

I'm trying to write the Lagrangian dual to the following problem \begin{align*} (P) \quad \min\;&\text{Trace}(CG)\\ \text{s.t.}\;&G \succcurlyeq 0\\ & G_{i,i}=I_d (i=1,..,M+1)\end{align*} ...
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7 views

If $d^k$ is a descent direction for $f(x^k)$, then $\nabla f(x^{k+1})^t d^k=0$, where $x^k+1=x^k+\alpha^k d^k$

I have solved this problem but got something slightly different. My professor is notorious for his typos in half of the examples, so I was wondering if this is just another typo or if I got it wrong. ...
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61 views

Convert a nonconvex function to convex function

I have a image $I: \Omega \to \Bbb R$. It is separated into 2 non-overlapping region: $D$ and $\Omega \setminus D$ Each point $x$ in the image $I$, the $\phi$ function is defined as: $$\phi(x)= ...
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23 views

Justifying the “Dual feasibility”, one of the Karush-Kuhn-Tucker conditions

I am having difficulty of interpreting the KKT conditions in a general setting where we have $M$ equality and $N$ inequality constraints defined as: Minimize $f(x)$ subject to $g_i(x) \leq 0 , h_j(x) ...
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2answers
34 views

Quite rare kind of proof of convexity for a quadratic function!

Excuse me all of you in advance. I got this problem as an assignment but I am not really good doing proofs! If $f(x)=\frac12x^TQx+b^Tx+a$ is quadratic in $n$ variables, where $Q$ is symmetric. Show ...
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42 views

Why does $\frac{\textbf{g}^T\textbf{d}}{\textbf{d}^T\textbf{H}\textbf{d}}$ give the maximum of function $\mathcal{D}(\textbf{x}+\lambda\textbf{d})$

Let say I have to find the value $\lambda^*$, that maximizes the following quantity: $$\lambda^*=\underset{\lambda\in \Lambda}{\text{arg max}}\;\;\mathcal{D}(\textbf{x}+\lambda\textbf{d}),$$ ...
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29 views

What is the approach to solve simple constrained optimization when first order condition $\nabla f = 0$ yields solution outside of domain

I wish to solve the problem min $ f(x_1, x_2) = x_1^2 - x_1 + x_2 + x_1x_2$ subj $x_1\ge 0, x_2 \ge 0$ We find $\nabla f = [2x_1 - 1 + x_2, 1+x_1] = 0$ yields $x_1 = -1, x_2 = 3$ which is outside ...
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11 views

Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$

I have been stuck at this problem for a while :( Given $\mathbf{A}\in\mathbb{S}^{p\times p}, \mathbf{A}\ge 0,\mathbf{A} \text{ symmetric}, \mathbf{b}\in\mathbb{R}^n,\mathbf{c}_i\in\mathbb{R}^p\forall ...
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22 views

Why steepest descent gives a wrong direction search?

I have to minimize the function $ƒ(x_1,x_2)=(x_1-1)^2+x_2^3-x_1x_2$. The initial point is $[1,1]^T$. The gradient of this function is $∇ƒ(x_1,x_2)=[2(x_1-1)-x_2,3x_2^2-x1]$. This gradient evaluated ...
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61 views

Karush-Kuhn-Tucker conditions for non-linear optimalization

I have the following problem: solve the local conditions (KKT) and find ALL optimal solutions: $$\min f(x,y)$$ subject to $$g(x,y)\le 0$$ $$x\geq0, y\in\mathbb{R}$$ I have some questions to this ...
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1answer
40 views

(half) hyperboloid least squares problem

I have five equations as follows, , where i = 1, 2, 3, 4, 5 and only (x, y, z) are unknown. The five equations above are half-side hyperboloids. It could be seen as . I want to find the solution (x, ...
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1answer
104 views

How to project gradient vector to subspace defined by linear constraints

I have the following set of linear constraints: $$\begin {align}\textbf{y}^T\textbf {x} &= 0 \\ \textbf {0} &\leq\textbf {x} \leq C\cdot\textbf {1},\end {align}$$ where $\textbf {y} \in ...
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30 views

what kind of optimization is this?

I have an optimization problem that looks like this: \begin{array}{cc} min & x'\varSigma^{2}x+k^{2}e'e-2ke'\varSigma x\\ s.t. & x'\varSigma x=ke'x\\ & ...
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1answer
49 views

How to find Parameters in nonlinear Regression Model?

I have a nonlinear Regression Model with eleven observations of $x,y$. How do I find the parameters $a,b,c,d$ of the model: $ y=f(x)=a + b \sin cx e^{dx}$ by using the function: $$\Phi(a, b, c, ...
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17 views

How to find parameters from logistic equation

I have an function and assume that that is convex function. I want to use gradient decent to find parameters in that equation. Could you suggest to me the way to do it. Thanks. This is my function ...
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20 views

Optimization with Integral Inequality constraint and nonnegativity conditions

Trying to solve this: $$\min TC(A,a,q)= \int_M f(A,a,q)\,dx\, dy$$ $$s.t.$$ $$a\le\int_M g_i(A,q)\,dx\,dy$$ $$q\le \text{constant}$$ $$A,a,q\ge0$$ $(x,y)$ is omitted in $A(x,y), a(x,y), q(x,y)$ ...
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20 views

Plotting Non Linear Programming functions

I define these functions in Matlab: ...
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1answer
41 views

How to determine the optimal step size in a quadratic function optimization

I have the following optimization problem: $$\underset{\alpha\in\mathbb{R}}{\text{min}}:\;\;f(\textbf{x}+\alpha\textbf{d})$$ $$\text{subject to}:\;\;0\leq\alpha\leq \alpha_{max},$$ where ...
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1answer
78 views

How do I convert a constraint with a product of two integer variables to a linear constraint?

I have a constraint of the form: $$\theta \leq a_1x_1 + a_2x_2 + a_3x_1x_2$$ where, $x_1$ and $x_2$ are integer variables with ranges $x_1 \in \{0, m\}$ and $x_2 \in \{0, n\}$. I would want to ...
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3answers
84 views

ADMM formalization

I found lots of examples of ADMM formalization of equality constraint problems (all with single constraint). I am wondering how to generalize it for multiple constraints with mix of equality and ...
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32 views

Do quasi-Newton methods check the second-order optimality condition?

I have a practical question about quasi-Newton methods. In quasi-Newton methods, Hessian matrix is approximated. It seems to be impossible for them to check the second optimality condition. In ...
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31 views

How to find optimized value of two variables

I have two variables: $\kappa_y$ and $\kappa_x$ And three functions: $M_y$=$M_y$($\kappa_y$, $\kappa_x$) $M_x$=$M_x$($\kappa_y$, $\kappa_x$) $F_z$=$F_z$($\kappa_y$, $\kappa_x$) All these three ...
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2answers
51 views

Why a convex cone cannot have more than one extreme point?

The way I define an extreme point is : A point which cannot be defined as a convex combination of two distinct points. I'm not able to extend this and show why a convex cone cannot have more than ...
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33 views

Why Was Backprop Invented?

I'm currently researching artificial neural networks and I keep wondering why do we use "backpropagation" to train a neural network. An ANN is basically just a very large and complex function ...
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43 views

Lagrange multiplier method

Question 1: Could somebody please refer me to an introduction to Lagrange multipliers which is easy to read but still in full generality? Question 2: I am interested in particular in the following ...
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1answer
29 views

How can I compute fast the minimum of a linear plus Kulback-Leibler on the unit simplex?

Given $a, x^0 \in \mathbb{R}^n$ I wish to compute $$\min_{x \in \Delta_n} a^t x + \sum_{i=1}^n x_i\log(x_i/x^0_i) - x_i +x^0_i $$ where $\Delta_n$ is the unit simplex $\{x \in \mathbb{R}^n \mid ...
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1answer
38 views

Gauss-Newton Non-Linear Squares Optimisation

I doubt this is solvable at all, but I thought I will give a try. Essentially I am trying to extend Gauss-Newton algorithm to 2nd Taylor term. ...
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22 views

Book on duallity and sensitivity in nonlinear optimization

I am looking for a recommended book on duallity and sensitivity in nonlinear optimization, as duallity and sensitivity is a well studied topic in LP , I am struggeling to find books in this subject ...
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23 views

Maximize a concave function under nonconvex constraints

I have to maximize the rate, which is a concave function, under certain constraints, where one of them is not convex; My optimization problem is: $\max_{\mathbf{P}_{2,n}} \frac{B}{L} \sum_{k=1}^L ...
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1answer
20 views

How do I know that method of steepest descent works?

Here is the definition of the method of steepest descent given in the book "The mathematics of nonlinear programming" by Peressini. Suppose $f(x)$ is a function with continuous partial derivatives on ...
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29 views

Global stochastic maximization of a multi-parameter function

I have a function $F:\mathbb{R}^n\to[0,1]$ such that $$ F(\lambda) = \mathbb{E}_x[f(\lambda;x)] = \int f(\lambda;x)\mu(x)dx,$$ and I want to find $\tilde\lambda$ that maximizes F, i.e. ...
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35 views

Solving Nonconvex Programming using Genetic Algorithm

Consider such a programming problem, \begin{equation} min \| \Psi(\boldsymbol{x}) \|_p \quad s.t. \quad \boldsymbol{Ax} = \boldsymbol{b} \end{equation} Where $\boldsymbol{x} \in \mathcal{R}^{n}$, ...
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1answer
52 views

How to maximize $\sum\limits_{i=1}^n u_iln(x_i)$?

How to maximize this? $$ \sum\limits_{i=1}^n u_iln(x_i), $$ where $u_i,x_i$ are real numbers, $n$ is a positive integer, $0 \leq u_i \leq 1, 0 < x_i < 1, \sum\limits_{i=1}^n u_i = 1, ...
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1answer
81 views

No clear analytic method to prove unique maximum? ($2^{-x}+2^{-1/x}$)

Prove that $f(x) = 2^{-x}+2^{-1/x}$ has the unique local maximum $(1,1)$ for $x>0$. Do not use computer software. Proving that $(1,1)$ is a maximum is easy, but I'm having trouble with the ...
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32 views

How to prove a solution is indeed a constrained minimum?

I'm reading the following example on Heath's Scientific Computing (page 266, second edition if anyone has it). "Minimize $f(x_1,x_2)=2\pi x_1(x_1+x_2)$ subject to $g(x_1,x_2)=\pi x^2_1x_2-V$" ...
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22 views

Linear programming (or possibly nonlinear) formulation

The problem is like this; The construction company is considering erecting three office buildings. The time required to complete each of them and the number of workers required required to be on the ...
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129 views

Epsilon constraint method - Pareto optimal solution representation

There's a course that I do remotely and I have a homework question which I have no idea how to answer. I did look up a lot in google and did not find any good examples - only loads of information and ...
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1answer
44 views

Non-uniqueness of worst-case (max-min or min-max) optimization

I have a worst-case optimization problem, where i want to maximize the minimum from the uncertainty set (uncertainty is given as an ensemble of 100 realizations, so an ensemble based approach). It is ...
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10 views

Can this equation have an explicit solution?

Given $n > 0$, $0 \leq i \leq n$ is an integer, $D = diag(d_1, \dots, d_n)$ is positive definite, $e_i$ is the $i$th column of a $n \times n$ identity matrix, $u \in R^n$ such that $B = D + u * ...
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1answer
81 views

How do we plot nonlinear differential equations

If this is not nonlinear I apologize, I'm still learning differential equations. I am attempting to make a stream plot of a predator-prey model of eccentric closed curves by using the following ...
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36 views

Optimization on fixed sum

Consider this following scenario. Suppose I have $N$ cents, and I want to dispatch these money to $n$ people, each got $x_i$ cents. In order to simplify this problem, we assume the cents are ...
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20 views

Dual norm of quasi norms

The dual norm $\Omega^*$ of the norm $\Omega$ is defined for any vector $\mathbf{z} \in \mathrm{R}^N$ by \begin{equation} \Omega^*:= \underset{\mathbf{x} \in \mathrm{R}^N}{max } \quad \mathbf{z}^{T} ...
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1answer
27 views

How to optimize these parameter

How to optimize the following respect to lambda1 and lambda2: $\sum_{i} f(i) * log(\lambda_1 g(i) + \lambda_2 h(i))$ f(i), g(i), h(i) are known funtions Find lambda1 and lambda2 that satisfy ...
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1answer
44 views

Necessary Conditions for Saddle Value point

This questions is from the Kuhn-Tucker paper "Nonlinear Programming" in Section 2 Lemma 1. I don't understand how those conditions are necessary for a saddle point. I always thought that a saddle ...
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43 views

Non-convex function with global minimum [duplicate]

I am working on a complicated objective function which I suppose is not convex. But when I use a global optimization tool that can find all its local minimums, it will always converge to the same ...