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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Select positions for strongest defense given probability(position, target) scores. [closed]

In a game of tower defense, I want to place archers to optimize survival time. I have ~10 towers, and I am allowed one archer per tower. The towers have 50 to 300 vantage points each. Once an archer ...
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72 views

is there a closed form solution to this continuous optimization problem?

Consider the function \begin{eqnarray} \max_{t_1,\ldots,t_p \ge 0} V(p) & = & \sum_{i=1}^p [- \alpha t_i - \beta e^{\rho - \delta^{i-1}\theta} \prod_{k=1}^i t_k^{-\delta^{i-k}\Omega}]. ...
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41 views

add a point, close to one but distant to the others

I am given a finite set of geometric points in three dimensions. I want to add another point $A$, so that it's close to a certain point $P$ (that is $d(P, A)\lt k$, where k is an arbitrary constant), ...
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3answers
730 views

Batch vs incremental gradient descent

I am studying Machine Learning, but I believe you guys should be able to help me with this! Basically, we have given a set of training data $\{(x_1,y_1), x(x_2,y_2), ..., (x_n, y_n)\}$, and we need ...
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1answer
736 views

Local extreme value & saddle point: multi variable calculus

I am asked to find all local extreme values & saddle points of $$f(x,y) = 2x^2 + y^2 - xy - 7y + 8$$ $$f_x(x, y) = 4x-y, \qquad f_y(x,y) = 2y-x-7$$ $$f_x(x,y) = 0, \qquad y = 4x$$ $$f_y(x,y) ...
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2answers
3k views

Is it possible to calculate weights of a portfolio with negative values?

Sorry in advance if this question is either too basic or really dumb, but I've been researching this and am a bit confused. I'm trying to help my niece with a question she has and the gist of it is ...
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2answers
183 views

Compressed sensing, approximately sparse, Power law

An x in $\mathbb{R}^n$ is said to be sparse if many of it's coefficients are zeroes. x is said to be compressible(approximately sparse) if many of its coefficients are close to zero.ie Let ...
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157 views

How to find the best (integer) polynomial equation?

This is a continuation of How to find curve equation from data? I asked earlier. I am looking for both a formula and the method to find the best (integer) polynomial that fits my data. I still ...
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3answers
38 views

Discontinuity phenomenon in polynomial optimization

Let $Q(x,y)$ be a polynomial in the two variables $x$ and $y$, with real coefficients. Let $a<b$; for any fixed $x$ the polynomial $Q(x,.)$ has a minimum $m(x)$ when $y$ varies in $[a,b]$. We know ...
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1answer
308 views

Find equation of line such that area formed by line & positive coordinate axis is minimal

Find equation of line passing through $(20,12)$ such that the area of the triangle formed by the line and the positive axis is smallest possible. Also: $\frac{x}{a}+\frac{x}{b}=1$ where $a, b$ are ...
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1answer
176 views

Solving linear programming problem with global opt method

why not solve a linear programming problem with a global opt method, or a local search method as SQP or Newton methods? I am writting a solver facing linear and non linear problems, and I wonder ...
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1k views

generalized Rayleigh Quotient

If $M$ is positive definite, $H$ is self-adjoint. Now consider the minimization problem:$$\min_{x\neq 0}\frac{(x,Hx)}{(x,Mx)}.$$ Note that this functional is homogeneous of degree 0. So we can just ...
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79 views

Interval Algorithm for Gradient Descent Method

Are there any references discussing an interval algorithm for the vanilla gradient descent method given a function $f \colon \mathbb{R}^n \to \mathbb{R}$? Edit: In particular, I am searching for an ...
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2answers
152 views

Dynamic Optimization - Infinite dimensional spaces - Reference request

Respected community members, I am currently reading the book "recursive macroeconomic theory" by Sargent and Ljungqvist. While reading this book I have realized that I do not always fully understand ...
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1answer
69 views

Upper bound of an expressions with many variables

Assume $0 < p_1 \le p_2 \le \dots \le p_{2k}$. I am looking for a (preferably tight) upper bound for the following expression: $$ \frac ...
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1answer
82 views

Maximum of an expression with four variables

Assume $0 < p_1 \le p_2\le p_3 \le p_4$. What is the maximum of the following expression? $$ \frac{\left(p_1+p_4\right)\left(p_2+p_3\right)}{\left(p_1+p_3\right)\left(p_2+p_4\right)} $$ Is that ...
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1answer
129 views

Optimization for large scale linear problem with equality constraint

Given the wide range of optimization methods, which is the appropriate method to use? I am thinking of using either linear programming (interior-point methods) or augmented Lagrangian methods. Which ...
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78 views

Portfolio optimization - problem with a proof

I'm trying to proof Proposition 1 in this Paper about Markowitz Portfolio Opitimization on page 6/7 but I can't figure out how to do this. The author wrote "The proof of Proposition 1 can be found ...
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219 views

Calculating the Dual Problem to a Minimisation Problem - confusion

Let $A$ be an $m$x$n$ real matrix. Let QP denote the problem: minimise $f(x,y) = \frac12 x^Tx$ such that $x \ge 0, y\ge 0$, and $Ax-y=b$. I want to prove that the dual of this problem QP* is: ...
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1answer
88 views

Existence theorems for problems with free endpoints?

It is well known that the problem of minimizing $$ J[y] = \int_{0}^{1} \sqrt{y(x)^2 + \dot{y}(x)^2} dx $$ with $y \in C^2[0,1]$ and $y(0) = 1$ and $y(1) = 0$ has no solutions. However, if we remove ...
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1answer
205 views

Nash Equlibria and Maximin Strategies

Consider the following bimatrix game $(2,6)\ \ \ (4,2) \\ (6,0) \ \ \ (0,4) $ I have been asked to compute all equilibria of this game, as well as the maximin strategies for both players. Now I used ...
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71 views

Petri net analysis (attainability)

how to analyse safe petri net for attainability? (i need algorithm) I have an oriented multigraph $\mathbb{G}$. $A$ - adjacency matrix. $m$ - the count of input elements. $n$ - the count of ...
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219 views

Find $y$ to minimize $\sum (x_i - y)^2$

I have a finite set of numbers $X$. I want to minimize the following expression by finding the appropriate value for y: $$\sum\limits_{i=1}^n (x_i - y)^2$$
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159 views

Linear Programming Duality (Basic optimization)

Suppose that $A$ is an $m\times n$ matrix, $D$ is a $p\times n$ matrix, $b$ is an $m$-vector, and $d$ is a $p$-vector. Prove that there does not exist $n$-vector $x$ satisfying $$Ax \geq b, Dx \leq ...
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48 views

Maintaining the line with the 2D iterands

Suppose a linear system is given $$AX=B,$$ where $A\in\mathbb{R}^{n\times n}$ is a symmetric strictly diagonal matrix, and $X, B\in\mathbb{R}^{n\times 2}$. Therefore, the 2D Jacobi iterative solver is ...
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1k views

Find the maximum of $f(x)=x^{1/x}$

Find the maximum of the function $$f(x)=x^{1/x}$$ and the value of $x$ which gives the maximum value?
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459 views

Efficient Cholesky decomposition of inverse matrix

I want to generate random numbers from a multivariate normal distribution in Matlab. Normally, this is done like: $w = \overline{w} + \text{chol}(\Sigma) \cdot \vec{l}$ But in my case I don't know ...
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2answers
660 views

Largest circle between $y=x^n$ and $y=\sqrt[n]{x}$

Something I have been wondering about for a while. Let us look at the area between $x^n$ and $\sqrt[n]{x}$ when $x\in [0,1]$. Where $n$ is a positive integer. Below is an image. With a given n, how ...
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108 views

How to solve $\max_{f}\int_{0}^{\infty}f\left(x\right)dx$ subject to $\int_{0}^{\infty}xf\left(x\right)dx=x_{0}$?

How to find $\max_{f}\int_{0}^{\infty}f\left(x\right)dx$ subject to $\int_{0}^{\infty}xf\left(x\right)dx=x_{0}$, where $f$ is a function and $x_{0}$ is a constant?
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1answer
438 views

Euler lagrange assumptions

I have a question related to these two posts: (1) Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising and (2) When the Euler Lagrange equation simplifies to zero Background ...
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1answer
101 views

Is this function convex?

I have a model - function of two vectors $A$ and $B$. I have data that I want to fit to the model and find the model's parameters. The function needs to be convex to find the parameters using ...
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1answer
204 views

minimum number of vertices for a specific graph

Today I saw this problem: Find the smallest $n\ge 5$ such that there exists a simple graph on $n$ vertices such that any two adjacent vertices have no common neighbours, and any two non-adjacent ...
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157 views

Shortest Fence in a Quarter-Circle Pasture

suppose that there is following problem A farmer needs to divide equally a quarter-circle pasture between two of his cows. He seeks to build a shortest straight line fence. Which of the three ways ...
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63 views

Is there a mathematical approach to optimizing given these conditions?

The idea is you have a set of things that you know and a problem to solve. Sometimes you think you know enough but are wrong. Sometimes you think you don't know enough but are wrong. In one case ...
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36 views

How to discuss the maximum Area of Internal rectangular in an irregular region?

How to discuss the maximum Area of Internal rectangular in an irregular region? such as Fan-shape,or the region....
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1answer
2k views

Matlab Trust-region-reflective algorithm warning

I am very new to matlab and trying to solve portfolio optimization problem (minimizing the variance) using quadprog: ...
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125 views

No identical rectangles in a matrix

I have a matrix of dimensions N x M. Every cell has an integer. Now, I want for every 'rectangle', to verify that all its corners are not the same. Example: This matrix is fine: This matrix is not: ...
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1answer
146 views

Correctly adding constraints to Ax=b

I have a function of the form $$ E(\mathbf{x})=E_1(\mathbf{x})+E_2(\mathbf{x}) =\sum_i\|\ldots\|^2+\sum_j\|\ldots\|^2 $$ and want to solve the optimization problem $$ ...
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1answer
83 views

Least squares and (non-)linearity of parameters

I have a question about least squares and about what happens, if the function that we minimize, $E(P)$, is not linear in its parameters $P$. Assume we want to minimize a function (the exact terms are ...
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196 views

$\beta_k$ for Conjugate Gradient Method

I followed the derivation for the Conjugate Gradient method from the documents shared below http://en.wikipedia.org/wiki/Conjugate_gradient_method ...
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2answers
332 views

Rate of convergence of a sequence in $\mathbb{R}$ and Big O notation

From Wikipedia $f(x) = O(g(x))$ if and only if there exists a positive real number $M$ and a real number $x_0$ such that $|f(x)| \le \; M |g(x)|\mbox{ for all }x>x_0$. Also from Wikipedia ...
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1answer
213 views

How to optimize entropy under linear constraints?

My problem is quite cumbersome. In general, it can be modelled as a non-linear programming problem, with linear constraints and non-linear objective function. The objective function is conditional ...
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38 views

Prove a ordering computation complexity theorem.

If I want to find out the order to comparing $n$($n$ is very big like an million) apples(weight different for each apple) with no other information, but no limition of apples putting on the balance at ...
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150 views

Linear Program feasibility

Let $A$ be an $m \times n $ matrix, $b \in \mathbb{R}^n$, and consider the linear program $$\max\{ 0^Tx: Ax = b, x \ge 0\},$$ and its dual $$\operatorname{min}\{y^Tb : y^TA \ge 0 \}.$$ Here $x \in ...
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maximum area of rectangle inscribed in a circle using geometric programming

need to find maximum area of rectangle that can be inscribed in a circle of radius r but need to use geometric programming of optimization to this for the maximum area the function is $ xy $ (if x ...
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252 views

Portfolio Optimization Problem Without Correlation Info

I received this interesting problem from a friend today: Assume that you are a portfolio manager with $10 million to allocate to hedge funds. The due diligence team has identified the following ...
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350 views

How shall I understand this simple example of No Free Lunch theorem?

I have trouble in understanding a simple example following No Free Lunch theorem in James Spall's Introduction to stochastic search and optimization: My understanding is that a cost function is a ...
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1answer
93 views

Existence of a general-purpose (almost) universal optimization strategy

From Wikipedia about interpretations of no free lunch theorem A conventional, but not entirely accurate, interpretation of the NFL results is that "a general-purpose universal optimization ...
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284 views

Is conditional entropy a convex function?

A conditional entropy can be expressed in the following way, $H_{V_t}(V_s) = -\sum_{s,t}p(s,t)\log{p_t(s)} = -\sum_{s,t}p(s,t)\log{\frac{p(s,t)}{\sum_{s'}{p(s',t)}}}$ $s$ and $t$ are defined ...
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302 views

Understanding no free lunch theorem

From Wikipedia: $Y^X$ is the set of all objective functions $f$:$X$→$Y$, where $X$ is a finite solution space and $Y$ is a finite poset. The set of all permutations of $X$ is $J$. A random ...