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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Minimizing a function in Mathematica

Edit: I simplified the function using $\textbf{Simplify[...]}$ How can I minimize this function of $x$, where $l$ is a positive constant? $$\frac{1}{2} \sqrt{\frac{x}{l}+\frac{l}{x}+4 x^2-2}$$ ...
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Solving nonlinear system of ODEs

I have the following system of differential equations: $$ \begin{cases} \frac{dx}{dt} = (1 - y) x - 0.4 xu \\ \frac{dy}{dt} = (x - 1)y - 0.2yu \\ \psi_1' = - \frac{dH}{dx} = (-1 + 0.4u)\psi_1 + y ...
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Constrained nonlinear optimizacion

I have encountered this optimization problem while trying to implement the method proposed in this very interesting paper: http://www.mae.cuhk.edu.hk/~cwang/pubs/JCISERealTimeSkeleton.pdf the ...
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41 views

What kind of an optimisation problem am I dealing with? [on hold]

I have a connected graph made up of $x$ vertices. Each vertex has a probability $p$. I want to determine the total probability in traversing as many vertices as possible, Edges have a certain cost to ...
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22 views

What does coordinate descent actually do?

We've done a bunch of theoretical stuff in my optimization class, but basically no time for the actual implementation details. I'm trying to get an understanding of coordinate descent, which if I'm ...
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Linear optimization w/ linear and non-linear inequality constraints

Given dependent variables $Q_i$ and independent variables $x_i$, $y_i$, $z_i$ where $i=1,⋯, N $ which are related via the following system of N linear equations with parameters $P_1$, $P_2$ and ...
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If $2000 m^{2}$ of material is used to to construct a box…,then what is the largest possible volume of the box?

If $2000 m^{2}$ of material is used to to construct a rectangular box with a square base and an open top,then what is the largest possible volume of the box? I put an equation for the volume : $V = ...
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25 views

Removing a max function in the constraints

Can the following problem be transformed into a linear programming problem: Find $x_1,..,x_N$ which maximizes the objective function $$\sum_{i=1}^{N}x_{i}\sum_{j=1}^{n_{i}}c_{ij}$$ subject to the ...
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12 views

MINLP optimization with matlab reaching different solutions every run

I have written a program for optimizing a set of generators. I have hourly price and cost data and need to figure out when a generator should run or just stay off. I describe the problem in more ...
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matlab MINLP optimization with ga

I have written a program for optimizing a set of generators. I have hourly price and cost data and need to figure out when a generator should run or just stay off. There are additional constraints but ...
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discrete nonlinear convex optimization relaxation over a dense set

Be a discrete nonlinear convex optimization problem $P$ \begin{align} \underset{x\in \mathrm{C}^n}{\mathrm{min}} \ \ \ f(x) \\ Ax=b \\ c \leq x \leq d \end{align} $C$ is a dense in $F$. Is solving ...
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Optimal value of decision variable leads to inconsistency

$\epsilon$ is a random variable with support in $(0.8,0.95)$ and pdf $f(\epsilon)$. The following equation arises out of a business problem: $ENP=800*A*E(\epsilon)+ 9000 - ...
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25 views

Minimization using logarithmic barrier function

I'm thinking of the quadratic problem(QP) \begin{align} &\underset{x\in \mathrm{R}^n}{\mathrm{Minimize}}\ \ \ \frac{1}{2}x^\top{}Qx + f^\top{}x\\ &\mathrm{subject\ to}\ \ \ \ a_ix \leq b_i\ ...
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23 views

Point spectrum of a nonlinear operator on finite dimensional space

Given a nonlinear operator $T$ mapping $\mathbb R^n$ into itself, are there any known conditions on $T$ ensuring that the number of points in its point spectrum is upper bounded by the dimension $n$?
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16 views

Optimal time control for the system of two non-linear ODE

I have the following system of two non-linear ODE with one control variable (modified model of Lotka-Volterra): Here is $\alpha, \beta, \gamma, \delta$ - some constants, $u$ - control variable. ...
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74 views

A binary min-max optimization problem

I encountered a very special optimization problem for a practical application. We have a variable $$\mathbf{s}=(s_1,s_2,s_3, s_4)^T$$, where $s_i$ can only take $1$ or $-1$, and we also have a ...
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direct connection between gradient descent and follow the (perturbed) leader algorithm or weighted majority? [migrated]

Is there a direct conversion between gradient descent ([1], Alg 1 ) and any of the following algorithms? 1) Weighted Majority: http://onlineprediction.net/?n=Main.WeightedMajorityAlgorithm 2) ...
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13 views

Polar cones' property [duplicate]

I am trying to prove: $A \subseteq B \implies B^\circ \subseteq A^\circ$ where $A^\circ$ is polar cone of $A$ ($A$ convex cone) and $B^\circ$ is polar cone of $B$ ($B$ convex cone)
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28 views

What does 'the level set is bounded' exactly want to tell?

'The level set is bounded.' occurs in many theorems and other places. I think I can understand the definition of 'level set' but I don't know what does 'it's bounded' want to tell me exactly in ...
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7 views

Finding a Sparse Binary Vector Which Satisfies a Vector Ordering

I have a research problem that I have reduced to the following problem which seems to be hard: Find $\mathbf{x}\in \mathbb{B}^N$ $(\mbox{where }\mathbb{B}= \{0,1\})$ which is $K$-sparse and which ...
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2answers
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Local optimality of a KKT point.

Consider the problem \begin{equation} \min_x f(x)~~~{\rm s.t.}~~~ g_i(x)\leq 0,~~i=1,\dots,I, \end{equation} where $x$ is the optimization parameter vector, $f(x)$ is the objective function and ...
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2answers
46 views

How to get the perfect square for the following equation

The problem is defined as follows: $$ \min_X tr(X^T A X)-\alpha tr(X^T B) $$ I want to get the equal perfect square equation as that above, that is $$ \min_X \| X-C\|_F^2 $$ where $C$ is related to ...
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Is there a cost function for row equivalent matrices?

I am working on a minimization problem as follows: argmin$_x$ ||x-y||$_2$$^2$+$\lambda$||$\Psi$x||$_1$ where x and y are 2D or 3D complex arrays ||$\cdot$||$_1$ and ||$\cdot$||$_2$ are the L1 and L2 ...
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52 views

fantasy basketball model

i'm creating a fantasy basketball model (could be used in other games too) where we can project how well a player will do against another team even when the player hasn't played against a certain team ...
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27 views

Solving a Non-linear Multivariable System of equations

How would I go about solving a system of nonlinear equations where the highest degree is two? For example: $$f_1(x) = f_1(x_1, x_2,\dots, x_n) = 0,$$ $$f_2(x) = f_2(x_1, x_2,\dots, x_n) = 0,$$ ...
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29 views

Quadratic Program reformulation

I have the quadratic program $$\max\quad \mu^Tx+r_fx_0-\gamma \sum\limits_{i=1}^n |x_i-y_i|-\frac{\lambda}{2}x^TVx$$ $$\text{s.t. }\quad \mathbb{1}^Tx+x_0=1$$ where $\mu$, $r_f$, $\gamma$, $\lambda$, ...
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Newton's method: Is the change of parameter values between consecutive steps always decreasing?

Assume that I have a twice differentiable function $f(x)$ which I try to maximize with respect to $x$ (let's say $x$ is $k$-dimensional vector). When performing optimization via Newton algorithm, ...
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26 views

Are there any algorithms for simultaneous optimization of multiple objective functions?

I would like to minimize a set of similar objective functions $$f_\boldsymbol{s}(\boldsymbol{x}),$$ where $\boldsymbol{x} \in A \subseteq \mathbb{R}^M$ and the parameterization $\boldsymbol{s} \in S ...
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33 views

Roots of an equation using Maple

I am using Maple to find the roots of a non-linear equation in one variable. When I solve the equation, I get only 2 negative roots whereas if I plot the graph of the function, it also shows that the ...
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21 views

Direct multiple shooting (numerical optimal control)

please, Iam currently implementing direct multiple shooting method* and I need one simple but fundamental concept answered: When I want to provide not only objective funtion value (result of ODE ...
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1answer
28 views

Can you suggest a method to split a function in two parts such that the two parts may represent equitable magnitude contribution. [closed]

I was thinking about the sum of a even and odd function but the domain for the problem is positive real numbers only.
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24 views

SOCP or SDP optimization problem

I am studying an optimization problem \begin{equation} \mathbf{w}^* = \text{argmax} \sum_{d=1}^D \log \bigg( \frac{|\mathbf{f}_d^H\mathbf{w}|^2+c_1}{|\mathbf{f}_d^H\mathbf{w}|^2+c_2} \bigg)\\ \\ ...
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Sensitivity of polynomial global minimizers with respect to perturbations in the coefficients.

I'm trying to find the value of a global minimizers of a multivariate polynomial (4 variables) of high order numerically. The numerical values of the coefficients are coming from noisy measurements ...
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48 views

Which is the better way to optimize a function with 3 variables

I have an optimization function depends on 3 parameters a, b, and c. Which is the better way to optimize it? ...
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31 views

Newton's method for unconstrained optimization applied to a quartic function in R2

I am faced with the task of applying Newton's method to the following problem: $$ \text{min} ~~~~~ 8x_1x_2+\frac{1}{4}(x_1-x_2)^4 $$ where $x \in \mathbb{R}^2$. For clarification, the Newton method ...
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About constrained optimization

I've the following optimization problem:$$\min f(\theta_1,\theta_2)=\frac{a}{\cos\theta_1\cdot v_1}+\frac{b}{\cos\theta_2\cdot v_2}$$$$\operatorname{sub}\quad a\cdot\tan\theta_1+b\cdot\tan\theta_2=c$$ ...
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1answer
35 views

Why is the conjugate direction better than the negative of gradient, when minimizing a function

In gradient descent we minimize a function $f(\textbf{x})$, by using the update rule: $$\textbf{x}_{t+1} = \textbf{x}_t-\alpha\nabla f(\textbf{x}_t).$$ We also know, that at each iteration we have ...
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A weird optimization problem

I've the following optimization problem:$$\max f(R,z)=R^2(a+z)$$$$\operatorname{sub}\begin{cases}R^2+z^2=a^2\\0\le z \le a\end{cases}$$ Once solved it gives $z=a/3$, ... Consider now the ...
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Can a positive definite kernel produce a kernel matrix which has negative eigenvalues?

(1) I've read that a symmetric matrix is positive definite when its associated eigenvalues are all positive. I am learning SVM lately, and have come to know a $d$th-degree polynomial kernel ...
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42 views

What is the correct change of variables to yield convexity in this nonlinear optimization problem?

$$ \text{min. } x/y \\ \text{s.t. } 2\leq x \leq 3 \\ x^2+y/z\leq \sqrt{y} \\ x/y=z^2 \\ x,y,z\geq 0 $$ To transform this problem into a nonlinear convex optimization problem, both the objective ...
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Why test problems in convex optimization are mostly random?

Very often people who compare performance of different algorithms in convex optimization use randomly generated data. For instance, this often happens in compressed sensing and signal processing. Is ...
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Optimization problem in image processing

I recently heard of the interior point method in nonlinear optimization problems and was also told of its (likely) usefulness in image processing. MATLAB of course, has just a function for that called ...
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Linearization of non-linear model

How can I linearize this: $b{\rm{log_2}}(1+xy)$ where $b\in\{0,1\}$: binary integer variable $0\le x\le 3$: continuous variable (bounded) $y$ is a known constant greater than $0$.
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How to perform the linearization

Can anyone help me to linearize this following constraint: $b_1{\rm{log_2}}(1+x_1y_1)+b_2{\rm{log_2}}(1+x_2y_2)\le Az$ Here $b_1$ and $b_2$ are binary integer variables. $0\le\{x_1,x_2\}\le3$: ...
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36 views

Solve: tanh(x) = a*x + b - most efficient way

I work on DSP code, where some equations are of form: tanh(x) = a*x + b (tanh or other hyperbolic functions) Currently I use Newton-Raphson method. Is there a better/faster method of finding ...
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1answer
29 views

How to linearize this mixed-integer nonlinear constraint

Can someone please help me to linearize the following nonlinear/nonconvex constraint: $\sum\limits_{n=1}^Na_n\rm{log_2}(1+x_ny_n)\le M\delta$ Here $a_n \in\{0,1\}$, binary integer variable $0\le ...
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36 views

How to linearise this nonlinear constraint

I want to linearize or convexify this following constraint. Here $c_t$ is binary integer variables, $p_t$ are continuous variable which are bounded. $\gamma$ is a continuous variable. $h_t$ and $V$ ...
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What is a good optimization algorithm/tool for otimization on Partially Ordered set?

Actually I'm interested to minimize following kind of functions: $f: U \rightarrow V$ where: $U$ is a vector space and $V$ is a Ordered vector space, i mean Partially Ordered Vector space. ...
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23 views

Solving large non-linear polynomial equation system

I have a 2 order equation system of 7 unknowns. It is constructed as this: F1=0,F2=0,F3=0...F7=0 of which F1=f1*f2,F2=f3*f4... And f1=a1*p1+a2*p2+a3*p3+a4*p4+a5*p5+a6*p6+a7*p7 a1~a7 are known ...
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How to convexify a function

I encountered a problem in nonlinear programming, the model is written: max a s.t. a = bd + c(1-d) where a, b, c, d are positive variables. b and c are bounded, and 0<=d<=1. I am wondering if ...