Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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motivation for BFGS Hessian update rule

The BFGS method approximates Newton's method by replacing the Hessian of a function $f$ with an approximate Hessian $B_k$. At each iteration, the Hessian is improved using the formula in equation five ...
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51 views

Newton step for functions which takes matrix arguments

I want to minimize a function $f(X)$ which takes a matrix $X$ as an argument, i.e. $\min_X f(X)$. Using a descent method I start at step $k$ with feasible matrix $X^k$ and get to the next $X^{k+1}$ by ...
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70 views

Optimization problem with many inequality constraints

I have a maximization problem with 4 variables and six inequality constraints. This problem is already solved and I have access to its solution. However, I would like to solve it analytically (not ...
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Relating a Taylor-expansion to a maximization problem

Suppose a continuous and twice-differentiable function $f_a(x): [\underline{x}(a), \overline{x}(a)] \rightarrow \mathbb{R}$ has a Taylor-expansion around $x^*$ such that $f_a(x) = \text{const} - r a ...
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43 views

Solving an equation with multiple unknowns from different sets of natural numbers

Is it possible to solve an equation: a*x+b*y+....+c*z-n = 0 where {a, b,..,c, n} are natural numbers and {X, Y,...,Z} are different sets of natural numbers? Is it possible to find minima if there is ...
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Question about unique fixed point.

Suppose $f(x,y)\in C^{2}(R^n)$, $g=(g'_1,g'_2)'$, $g_1(y)=\arg\min_xf(x,y)$ and $g_2(x)=\arg\min_yf(x,y)$. $g$ has a fixed point at $(x_0,y_0)$. $h=(h'_1,h'_2)'$, $h_1(y)=x_0+\dot{g}_1(y_0)(y-y_0)$ ...
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49 views

What's wrong with my derivation of this intertemporal first order condition?

I'm reading a simple maximization problem that goes like this: \begin{align} \max E_0 \displaystyle\sum\limits_{t=0}^{\infty} \beta^t \frac{c_t^{1-\gamma}-1}{1-\gamma} \end{align} subject to these ...
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36 views

Finding an optimal set of weights for combining correlated classifiers

In order to combine classifiers that are correlated with one another, I would need to solve the following optimization problem: Find a vector $\mathbf{w}$ that minimizes $\mathbf{w}^T M \mathbf{w}$ ...
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50 views

What are some methods to compare two pointwise graphs, if one increases more than other?

What are some methods (other than slope and correlation) to compare two pointwise graphs, if one increases more than other? By pointwise, I mean scatter plots and the points are joined by segments. ...
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Does this problem have an optimal solution

The problem is as follows, $\max_{g(\cdot)} -\int_0^{g(0)}(g(0)-x)g(x)dx$ s.t. $g(\cdot)$ is from the class of continuous strictly decreasing functions on $[0,1]$ and $0<g(0)<g(1)<1$. ...
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Numerical Methods

Assuming I am given a Program which can calculate the value of a continuous, infinitely differntiable (we cannot calculate these derivatives), real, positive function of two real variables which has ...
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33 views

Convexity of Euler-Lagrange Equation

Is it necessary for euler equation of my functional to be convex for a solution to exist or is it sufficient that the functional is convex
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16 views

Confusion related to least squares

I was reading this paper where they have modeled the ys given some samples xs,ys as The paper states that the above optimization problem is equivalent to a least squares problem. I didn't get how ...
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1answer
56 views

Monotonically Increasing Mapping?

$\mathbf{h}_1, \mathbf{h}_2\in\mathbb{C}^{n}$ are given column vectors and $a>0$ is a given constant. Consider the matrix ...
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31 views

Making the Smallest Number of Mistakes Possible

I have the following problem. I have a set of $k$ labelled points, $\left\{\mathbf{x}_i, y_i\right\}_{i=1}^{k}$, where $\mathbf{x}_i\in \mathbb{R}^{2}$, and $y_i\in\left\{-1,1\right\}$. I want to ...
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how to solve this optimization problem with functions?

Suppose $\theta\in[0,1]$ and $s(\cdot)$ is strictly increasing in $\theta$ with $0\leq(0)<s(1)\leq1$ and $s(0)+s(1)>1$ How could I solve the program, $\max_{s(\cdot)\in ...
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Convex optimization with positive definite and symmetry constraints

I have this function $$f(X) = -\frac{1}{2}y^T\mathrm{X} y + b^T\mathrm{X}^{-1}b + \frac{1}{2}\log|\mathrm{X}|$$ where $\mathrm{X}$ is a matrix consisting of three components $\mathrm{X = A+B+C}$ where ...
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23 views

Convex optimization issues

I have to optimize a function $f(a,b,c_{ij})$ which consists of a terms like matrix $\mathrm{X = A + B + C}$ where $\mathrm{A}$ is a diagonal matrix with the diagonal elements equal to $a$. ...
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76 views

Issues in optimization with positive definite constraints

I have this function $f(\mathrm{X})$ such $\mathrm{X}$ is a positive definite matrix which is equal to $\mathrm{A+B+C}$. $\mathrm{A}$ is a diagonal matrix with variable $a$ on the diagonal elements. ...
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34 views

Issues with implementing Newton's method for optimization

I have this function $f(\mathrm{X})$ where $\mathrm{X}$ is a positive definite matrix. I am trying to optimize this function using Newton's method. So at any point $\mathrm{X}$', I will make a ...
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308 views

Max- Min Optimization problem

I am a noob in mathematic, so I would need your help in solving the optimization problem below \begin{array}{l} \max\limits_{\bf l} \min \left( \left| {\bf g}_1 {\bf Ml} \right|^2, \left| {\bf g}_2 ...
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29 views

Modelling rational power functions as cones for conic quadratic programming

It's quite easy to find documentation online that shows that conic quadratic programming can be used with functions that have ration powers, such as x^(3/2). example I can also find examples of these ...
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Convergence of Preconditioned Newton Method

Suppose we have an unconstrained minimization problem, such that the initial point does not lie on a subset of our space where the function is convex (some eigenvalues of the hessian,$H$, are positive ...
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Confusion related to derivative

What is the gradient of $||x + b||_1$ with respect to $x$? Here $x$ is the variable and $b$ is the constant.
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optimization of a non-differentiable, component-wise step function

I would like to estimate the (local) minimum of a function $c:R^N \mapsto R^+$ where: $c$ is only differentiable almost everywhere, there exists a component $j$, such that $\frac{\partial ...
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34 views

Confusion related to how the hessian was derived

I was reading this paper I didn't get how the hessian was derived in this paper Can anyone please explain?
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26 views

Confusion related to an optimization problem

I was reading this paper I didn't get this optimization part where they are using Newton's method As you can see when they are trying to find the Newton's direction, they are using coordinate ...
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79 views

A stochastic programming with a chance constraint

Let $X$ be a bounded positive variable with an unknown probability density function (PDF) and $f(X)$ be a differentiable positive function. $$\begin{align*} &\min/\max ...
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53 views

Von Neumann Entropy Inequality

Suppose $\rho_1$ and $\rho_2$ are density matrices (and thus Hermitian, positive semi-definite matrices) and $\hat{w}$ is the solution of the following optimization problem $$ \hat{w} = ...
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Optimization: How can I limit parameters variability?

I'm optimizing the normal of a 3D plane minimizing the re-projection error (with the difference of the intensity of some pixels in two images that corresponds of some points that lies on that plane). ...
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64 views

Optimal division of money between a loan and a down payment

I'm in the market for a car. I have no credit and was hoping to build some with a loan towards a car. I don't want to develop bad credit, so I plan to take out a loan that I know I can pay off with ...
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Is there an easy solution to this constrained discrete minimisation?

Given $\vec{a}$, $\vec{b}$, and $c$ I want to find a discrete combination $\vec{n}$ (i.e. a vector with non-negative integer elements) to $$\mathrm{minimise}\left(\vec{n}\cdot\vec{a}\right)$$ Under ...
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Minimizing an unknown system's output

Let we have an unknown system with two inputs and two outputs. inputs $x=[x_1 x_2]$ and outputs $y=[y_1 y_2 ]$ The system have the following properties $ y_1 = f_1(x_1,x_2)$ ; $y_2 = f_2(x_1,x_2) $ ...
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getting sign of LP solution variables

I have an LP where I'm only interested in the sign of some of the variables of an optimal solution. The value itself does not matter. Currently I'm using cplex to get an optimal solution and take the ...
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How would I answer this question? (Reworded)

George wants your help to work out how many of each type he should stock in order to maximise his profit. There are three types of Snackboxes: A, B and C. A and C both cost 5 to produce, and B cost 7 ...
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A basic question on optimization

Let $f : R^n -> R$ be a real-valued function and let $\bf{d}$ be a feasible direction at $x \in \Omega$. Now I want to understand the difference between the quantities $\frac{\partial f}{\partial ...
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111 views

Linear programming with countably “infinite variables” and “finite constraints”!

Is it possible to do a linear programming with countably "infinite variables" and "finite constraints"? If not, what do you purpose? (Example Link): Maximum and minimum of an integral under integral ...
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Summation notation [duplicate]

Does anyone have any suggestions as to how I would be able to formulate this problem using summation notation for those of you who are familiar with it? Hermione has been busy packing her bag with ...
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92 views

How to see that K-means objective is convex?

I'm trying to proof that the objective of the K-means clustering algorithm is non-convex. The objective is given as $J(U,Z) = \|X-UZ\|_F^2$, with $X \in\mathbb{R}^{m\times n}, U\in ...
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Chemical reactions and solutions of a constrained optimisation problem

I have to find a solution for this problem: given $N$ materials of density $\rho_k$, find the mixture of them giving a compound of density $\rho$. From a mathematical point of view, we have to find ...
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25 views

extremal points on a manifold intrinsiclly

I am wondering if there is a geometric object for real analytic manifolds that characterizes extremal points of the manifold intrinsically. For instance, suppose I live in the manifold, can I ...
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Minimizing values related to median?? [duplicate]

I have got function - non linear(I thk), and a set of variable S=[(x1,y1),(x2,y2)...]. The objective is to find the value for ...
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34 views

Approaches to fitting noisy oscillatory data?

I have observations $\hat{f}$ from data at points $\mathbf{x}=\{x_1,\ldots,x_N\}$, that is modeled as a known oscillatory form $f(k\ x)$ (for example, the sinc function), where $k$ controls the ...
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40 views

Directive on Dimensionality Reduction

I have a data set (24 data records) which is in $\mathbb{R}^{13}$ and I need to project it to a lower dimension (at least to $\mathbb{R}^{3}$). My objective of the dimensionality reduction is to ...
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Extrema of a vector norm under two inner-product constraints.

If $\langle\vec{A},\vec{V}\rangle=1\; ,\; \langle\vec{B},\vec{V}\rangle=c$, then: \begin{align} max\left \| \vec{V} \right \|_{1}=?\;\;\;min\left \| \vec{V} \right \|_{1}=? \end{align} Consider the ...
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Maximizing summation given a constraint

I am writing a software-based algorithm to calculate an optimal solution and I am completely stuck. I need to maximize the following summation with respect to x: $ \sum_{k=1}^n {a_k(1+x_k) \over ...
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Difference between two model fitting schemes

We have some experimental data, $x \mapsto \hat{f}$ and we're trying to fit a known model of the form $$f(x\ \left|\right.\ a_1, a_2, a_3, b_1, b_2, b_3) = a_1 F(b_1, x) + a_2 F(b_2, x) + a_3 F(b_3, ...
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29 views

Confusion related to convexity of a quadratic function

Lets say I have the following function of X $f(X) = (AX^TBX)$ I didn't get why matrices A and B need to be psd to make f(X) convex. Clarifications guys
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55 views

Curve fitting using the Lagrangian multiplier

I have obtained two curves from experimental work, say A and B. I am trying to find the lagrangian multiplier of these two curves without knowing any function / equation as I only have the coordinates ...
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550 views

How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...