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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Is $\mathbf{y}^*$ a local minimizer of $f(\mathbf{h}(\mathbf{y}))$?

Let $f(\mathbf{x})$ be a twice differentiable function, where $\mathbf{x} \in \mathbb{R}^n$. Let $\mathbf{x}^*$ be a local minimizer of $f(\mathbf{x})$. Consider a differentiable and invertible ...
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Does it make sense to use optimization algorithms(Like ACO) in weighing average to find weighs

I am going to use a DEM fusion method using simple weighing average,I am going to use 2 inputs to create my fusion function W1X1+W2X2/(w1+W2)=result this is a ...
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55 views

How to solve a matrix equation for a scalar?

Given matrices $Q, P \succeq 0$, a vector $q$, a real number $\gamma$. How can one solve the equation $ q^T (Q+\lambda P)^{-T}P(Q+\lambda P)^{-1} q = \gamma$ for the scalar $\lambda$ in an efficient ...
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57 views

Does the arithmetic mean minimize the sum of absolute values of deviations? [duplicate]

We have $x_1,x_2,\ldots,x_n \in \mathbb{R}$. I conjecture there to be a number $M \in \mathbb{R}$ such that for any $i=1,2,\ldots,n$ the quantity $$|x_i - M|$$ is as small as possible. How do you go ...
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61 views

Why does the Hamilton Jacobi Bellman Equation imply Pontryagin's Minimum Principle

I'm having difficulty understanding the proof that allows us to go from the Hamilton-Jacobi-Bellman equation to to the Pontryagin Min(Max) Principle. Lets consider $x(t)$ and $u(t)$ as real valued ...
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63 views

In a linear program, how to add a conditional bound to x?

I am working with a standard linear program: $$\text{min}\:\:f'x$$ $$s.t.\:\:Ax = b$$ $$x ≥ 0$$ Goal: I want to enforce all nonzero solutions $x_i\in$ x to be greater than or equal to a certain ...
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61 views

Optimizing Choice of Life Partner

In this link, Hannah Fry mentions that a mathematical argument has been made towards the probabilistically optimal strategy for picking someone to settle down with. The claim is as follows: If we ...
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How can I solve $y \in (N_X + \nabla f)(x)$ via projection?

I a aware that if I'm trying to solve for $x$ the problem $y \in [\lambda I + N_X](x)$ where $y$ is a known vector, and $N_X$ is the normal cone given by $N_X(x) = \{u : \langle u, x - y\rangle ...
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46 views

Exchanging $\lim$ and $\inf$?

Suppose we have a sequence of functions $f_n(x)$ that converge to a limiting function $f(x) = \lim_{n \to \infty} f_n(x)$ for $\forall x \in [a,b]$. I was wondering under what conditions the following ...
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23 views

derivative of a function of a vector

I'm not familiar with the derivative of a function of a vector. I know that $\frac{\partial\mathbf{x}^T\mathbf{x}}{\partial\mathbf{x}}=2\mathbf{x}$, where $\mathbf{x}$ is a n*1 vector. However, for ...
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Bracket minimization with a<b<c s.t. f(a)>0, f(b)=f(c)=0

Suppose I have a function f that is continuous (not necessarily unimodal) over real numbers, and three points a < b < c s.t. f(a)>0, f(b)=f(c)=0. I wonder whether this claim is right: There ...
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45 views

Non-linear system vs minimisation problem

If you have a non-linear system of equations which can be formally written as : \begin{equation} \begin{cases} F_1(\mathbf{x})=0\\ F_2(\mathbf{x})=0\\ \ \ \ \ \vdots\\ F_n(\mathbf{x})=0\\ \end{cases} ...
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56 views

Fraction of area covered by three circles

Take a square with edges of size $10$. Now take take three circles of radius $5$. Prove that you can't cover the square with these three circles. Find the maximum proportion of the area of the ...
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70 views

Find the subspace $S$ to maximize the function $J=\frac{Tr(P_{S}A)}{Tr(P_{S}B)}$

Let $A$ and $B$ be two synmetric matrices, $B$ is positive and $A$ is non-negtive. Surpose a $k$ dimentional space $S\subset R^{n}$ , and let $P_S$ be its orthogonal projection matrix. QUESTION: Find ...
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Maximising a net present value function

I am looking at an equation for profit derived from fishing operations. This is defined in terms of a bounded integral (with an upper bound of $+ \infty$), so it's a Laplace transform really. It gives ...
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47 views

Performing Second Derivative test on multivariate function

I have two functions $f=xy^2$ and $g=x^2+y^2$. When optimizing $xy^2$ on the circle $x^2+y^2=1$ I get 6 critical points but when I try to perform the second derivative test, it equals 0, meaning that ...
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How to use CVX to solve this problem?

I have a function in the variables $x_{kl};\ k,l=1\ldots,m$, $$\sum_{i=1}^n \sum_{j=1,j<j'}^{N_i}\left( b_{ij} b_{ij'}- \sum_{k,l=1}^{m}x_{kl}f_k(a_{ij})f_l(a_{ij'})\right)^2$$ where ...
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21 views

Minimizer of $\frac{\lambda}{2} \| \theta - \theta^{(k)}\| + \text{Loss}_\text{hinge}(y \theta \cdot x)$

How do you find the minimizer of: $$\min_{\theta \in \mathbb{R}^d}\left\{ \frac{\lambda}{2} \| \theta - \theta^{(k)}\|^2 + \text{Loss}_\text{hinge}(y \theta \cdot x)\right\}$$ if $\theta^{(k)} \in ...
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16 views

Find the $LDL^{T}$ factorization of $A$ when in the range of the positive definite

I am trying to find the $LDL^{T}$ factorization of the following matrix $$ A = \begin{bmatrix} 1 & b \\ b & 4 \end{bmatrix} $$ when $b$ is in the range of positive definiteness. I have ...
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24 views

Maximization of probability that all packets are successful simultaneously

I have packet streams $1...k$ and, streams with Prob(err) $p1...pk$. The $p$'s are consts $>0$. I'd like to maximize the probability all make it simultaneously while I'm allowing at most $N$ ...
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29 views

how to minimize $\sum_{ij}a_i a_j P_{ij}$

I want to calculate the minimum of $$\sum_{ij}a_i a_j P_{ij}$$ for different choices of $\{a_{i}\}$, under the conditions $$a_i=\pm1,i=1,2\ldots,n,$$ and $P_{ij}$ satisfies $$\forall i,j ,\quad ...
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21 views

Minimization over a symmetric matrix

I'd like to know what are possible methods to minimize over a symetric matrix R. Example: min $||AX -B||_2^2$ The minimization is over A, such that $A^T = A$, $A \in R^{3x3}$, $X \in R^{3x\alpha}$, ...
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28 views

Find Minimizer and Minimum Value for a Function

I am trying to work through some problems to find the minimizer and minimum value of a function. The book I am using doesn't have a clear cut example and I can't seem to find a good example online ...
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53 views

Explicit solution for a linear program with two constraints

This is not a homework problem, although it wouldn't surprise me if it happens to exist in a textbook somewhere. Is there an explicit solution for the linear program $$\max_x c^Tx ~~ s.t. \\ d^Tx = q ...
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101 views

Optimization over vector spaces. Generalized KKT.

I am looking for the extension of the theorem I found in the book by Luenberger called "Optimization by vector space methods." Here is the statement of that theorem from Luenberger: Generalized ...
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14 views

Minimise the probability of a variable being positive?

So the problem statement is that we are given that ($v^Ts$ is the variance) $$x \sim \mathcal{N}(v^Tc,v^Ts)$$ Where $c,s$ are constants and $v^Tv=1$. Show that minimising $\mathbb{P}(x>0)$ with ...
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Maximizing the volume of a cylinder with given area

Consider a right circular cylinder of radius $r$ and height $h$. It has volume $V=\pi r^2 h$ and area $A=2\pi r (r+h)$. We are to use Lagrange multipliers to prove the maximum volume with given ...
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42 views

Calculate the maximum area (maximum value)

TX farmer has 100 metres of fencing to use to make a rectangular enclosure for sheep as shown. He will use existing walls for two sides of the enclosure and leave an opening of 2 metres for a gate. ...
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58 views

To Calculate Maximum Volume by finding out Maximum value

Snacks will be provided in a box with a lid (made by removing squares from each corner of a rectangular piece of card and then folding up the sides) http://www.imageshack.com/a/img661/6094/fZUQXg.jpg ...
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Dual subgradient method - can we solve approximation of dual?

Consider the problem to minimize $f(x)$ under the constraints $x \leq b$ and $x \in X$. I Lagrange relax the constraint $x \leq b$ getting $L(x,u) = f(x) + u^t(x-b)$. When using the subgradient ...
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derivative wrt to a function

Suppose $\phi(x+V\Delta t)$-$aV{^2}\Delta t$ is a function to be maximized w.r.t the function V which is a function of (x), $a$ and $\Delta t$ being scalar constants. Assuming $\phi()$ is ...
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Formulation of Linear Programming problem?

I want to maximise the function: $$l(\beta,\sigma,\alpha) = -n\log(\sigma) - \frac{1}{\sigma} A(\alpha)\vert{\bf y}-{\bf X}\beta\vert,$$ where $\vert \cdot \vert $ represents the entry-wise absolute ...
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References for Worst-case max-min approach

I am looking for some good references for the max-min approach. Can someone kindly give me here? Specially when the max-min problem is solved by using a slack variable and additional constraints. I ...
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How to solve coupled nonlinear ODEs with a algebraic constraints?

Here is the problem I'm currently facing right now, I have set of ODEs which I can solve numerically given the initial condition. But I'm not sure how to go about giving algebraic solution with ...
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27 views

Maximum and minimum of this complex periodic function

I came up with this function by using fourier transform. My only problem is how to get the amplitude of this function. Im planning to get the difference between their maxima and minima. I get its ...
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Minimize : $\sqrt{(1+{1\over a})(1+{1\over b})}$ subject to $a+b=\lambda$.

Given positive real variables $a$ and $b$, find the minimum of $$f(a,b)=\sqrt{\left(1+{1\over a}\right)\left(1+{1\over b}\right)}$$ subject to $a+b=\lambda$ where $\lambda$ is a constant . [ISI ...
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106 views

Tangent portfolio weights without short sales?

Consider a mean-variance investor in a world with a risk-free asset. Let $R_f>0$ be the return of the risk-free asset, $\mathbb{E}(R_i)>R_f$ the expected return of the risky asset $i$ and ...
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How should I treat a linear relaxation to a rucksack optimization problem?

I am currently studying for an exam in optimization, and in one of the questions the following was mentioned: "The linear relaxation of the problem $(P)$: $z^*=\max ...
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38 views

Difference between two approaches to solving least squares

So if you're trying to find the least squares solution to minimize $||Ax - b||^2$, the normal approach is the pseudoinverse: $A^T Ax = A^T b$. If we define $r$ as the residual $(Ax - b)$, then we can ...
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Fit derivative to a set of points

Let's say I have a set of discrete values $X = {x_1, x_2, x_3, ..., x_n}$ from the sampling at a rate $f_s$ of a continuous function. I scale some values in $X$ (in a different manner for each one), ...
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What is the minimum of the following function in 8 variables, with the given constraints?

I need to minimize the following function: $f : \mathbb{R}^8 \to \mathbb{R}$, defined by \begin{align*}f(x_1, x_2, x_3, x_4, y_1, y_2, y_3, y_4)& =x_1x_2x_3x_4 + x_3 x_4 y_1 y_2 - x_2 x_3 y_1 y_4 ...
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27 views

Simplex Method: simplifying constraints

In my Computer Science class we've been exploring the Simplex Method and the applications it has with discovering optimal solutions. I've loved the challenge how much easier it makes finding solutions ...
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33 views

Optimizing for the minimum relative distance in a given situation?

I have primarily been working on this problem for quite some time now; the level of the problem is introductory calculus w/ optimization problems. The situation is as follows: Ship A sails due ...
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Proof the the Arithmetic-Harmonic Mean is expressible as the Geometric Mean

We define the Arithmetic-Harmonic mean of $a,b \in \mathbb{R_+}$ such that \begin{gather*} a_{n+1} = \frac{1}{2}(a_n + b_n) \\ b_{n+1} = \frac{2a_{n}b_{n}}{a_{n} + b_{n}} \end{gather*} Let us also ...
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Reducing a Knapsack-type problem to a known problem

The Quadratic Knapsack problem, introduced by Gallo, is an optimization problem in the following form: $max \sum_{i=1}^n{\sum_{j=1}^n{q_{ij}x_ix_j}}$ $s.t \sum_{i=1}^n{w_ix_i} \leq c$ $x \in \{0, ...
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Direct vs Indirect Learning Control

What is the difference between direct and indirect learning control? I found the following comments on direct and indirect control in this paper by Wang, Gao, and Doyle: "Survey on iterative learning ...
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60 views

Which optimization problem to use when I want to explain the concept of optimization to a layman?

I am looking some problem, which would be: Easy to understand Hard to solve intuitively Touch our everyday lives I am doing research in optimization using evolutionary computation. When people ...
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32 views

Showing that a function stays above one

This is a rather elementary problem: Consider the following function $$ f(x) = \frac{1}{1-e^{-x}}\Big[ 1 - \frac{x^2}{(e^{x}-1)^2} \Big] $$ defined on $[0,\infty)$. We have $f(0)=f(\infty)=1$ and ...
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When to use $argmin$ and $Argmin$?

What is the difference between the notations $argmin$ and $Argmin$ precisely? If I'm not mistaken one is used when the set of points attaining a minimum of a function has more than one point and the ...
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26 views

What is the optimal strategy for this made-up casino game?

Part 1: Let's say I walk into a casino with 100 dollars in my wallet. I sit down to play a game where my payoff or loss each round is aX dollars, where X is a continuous random variable uniformly ...