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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how to derive this equation?

How can I derive this? $\min_{d_m} \|Y - DX\|_2^2 = \min_{d_m} x_m^Tx_md_m^Td_m - 2R_mx_m$ where $R_m = Y - \sum_{i \neq m } d_ix_i^T$ $x_m $ is a vector represents a row in $X$
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What is a good optimization textbook for the theoretically inclined student looking for a rigorous and concise proof-based book?

I'm looking for an optimization book that is more like a classic pure math textbook without requiring any actual prior pure math courses. A book that puts focus on the theoretical aspects of ...
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Prove circle packing solution is optimal

Background: This is a follow on from this question of how to maximise the area of two non overlapping circles of arbitrary radii packed into a rectangle of arbitrary width and height. I proposed a ...
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35 views

How to approach a minmax problem?

Starting with a certain geometric problem, I have reached this function: $$R(s,t,u,v)=\max(s-u,s+u,t-v,t+v,sX+tY+u, tX-sY+v)$$ where $X\geq0$ and $Y\geq0$ are parameters. I have to find the minimum ...
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75 views

Optimization Calculus.. a box/shelter with sides missing..

I'm solving a problem involving calculus optimization. The problem is the following: "We plan to build a boxshaped shelter with no floor and one side open. (Hence we need a roof and three sides). The ...
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BFGS update formula

In this pdf : http://www.ing.unitn.it/~bertolaz/2-teaching/2011-2012/AA-2011-2012-OPTIM/lezioni/slides-mQN.pdf, in slide 46, the BFGS updata rule is given and is simplified to a second form. How did ...
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25 views

How to apply Sherman Morrison formula for rank 2 update?

For obtaining the inverse update in BFGS, Sherman-Morrison needs to be applied twice since it is a rank 2 update. But what does it mean to apply it twice?
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Minimum perimeter of a three-sided rectangular fence with given enclosed area

A three-sided fence is to be built next to a straight section of a river, which forms the fourth side of a rectangular region. The enclosed area is equal to 1800 ft^2. Find the minimum perimeter and ...
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22 views

Modelling problem

i have this problem and i have to model it in a boolean formula. Assuming that variables can have value 0 or 1 and V is OR and ∧ is AND. I have n boolean variables x1,x2......xn. i want a formula ...
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2answers
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When is $A^t+B^{1-t}$ for $t \in[0,1]$ maximized?

When is $A^t+B^{1-t}$ for $t \in[0,1]$ maximized. Suppose that $A,B \in \mathbb{R}^{+}$. This is very similar to convex combination but only in exponents.
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414 views

primal to dual solution conversion ??

i have an optimization problem $$\text{ maximize } z=3x+4y$$ $$\text{ such that: } x+y ≤ 450 \text{ and } 2x+y ≤ 600$$ the optimal solution to this problems comes to be $x=0$; $y=450$; $p=150$ ...
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36 views

Proving a point is a local minimum

I have a rather basic question. I have a function $f:R \rightarrow R$, and I want to show a point, $x^*$, is local minimum, i.e., $f(x^*+\delta) \geq f(x^*), \ \delta \to 0$. I can show that: $f(x^* ...
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How to solve this equation (may be with least squares)?

I have a system of linear equations in the following form. How can I solve it? $$\operatorname*{argmin}_{a,b} \sum_{i,j} \left( \left| X(i,j)-aY(i,j)\right|-b \right)^2$$ Where $X$ and $Y$ are ...
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2answers
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Find the maximum points of $f(x)=e^{-x}\sin^2(\pi x) \hspace{0.4cm},0<x<10$

Find the maximum points of $$f(x)=e^{-x}\sin^2(\pi x) \hspace{0.4cm},0<x<10$$ My calculations:I have calculated $f'(x)=\pi e^{-x}\sin(2\pi x)-e^{-x}\sin^2(\pi x)$ $f''(x)=e^{-x}\sin^2(\pi ...
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27 views

Talyor's Theorem in Nocedal's Numerical Optimization

Please kindly refer to the figure below. I understand that (2.4) is just another formulation of Mean Value Theorem and I understand its geometrical meaning in 1-D case. However, I do not know what ...
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maximum likelihood of a dirichlet prior

Suppose $\theta \sim D(\alpha)$ where $D$ denotes the Dirichlet distribution and $\alpha = (\alpha_1,\ldots,\alpha_K)$ its hyperparameter, in which case: $$p(\theta) = \frac{\Gamma(\sum_k ...
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Finding an optimal path for minimizing an integral.

Let $x,y$ be real numbers. Let the function $f(x,y)$ be real-entire in both $x$ and $y$. Thus $f(x,y)$ is a real-entire Taylor series in the variables $x,y$. How the find a non-intersecting path ...
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26 views

Is there a two-dimensional method to optimally allocate N sampling points on a continuous function with derivatives?

I am looking for a method to optimally allocate sampling points. I have read some papers on this topic that discuss one-dimensional allocation using chebyshev points, but I haven't found a good ...
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4answers
46 views

Is $\max\left(\frac{A}{B},\frac{C}{D}\right) \ge \frac{\max(A,C)}{\max(B,D)}$

Is $\max\left(\frac{A}{B},\frac{C}{D}\right) \ge \frac{\max(A,C)}{\max(B,D)}$? Given that $A,B,C,D>0$. What about $\frac{A}{B},\frac{C}{D}>1$. Is there a better bound for the left ...
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1answer
22 views

Logistic regression maximum likelihood derivation

the following equations are given: $\sum_{j=1}^c\hat{P}_j = 1$ $\sigma_i(\mathbf{z}; \theta) = \frac{exp(\mathbf{\theta}_i^T\mathbf{z})}{\sum_{j=1}^cexp(\mathbf{\theta}_j^T\mathbf{z})}$ $L = ...
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Prove that the partial derivatives of $(y-g_i+a\sum^n_{j=1} g_j)$ are positive

I have a function: $$\pi_i^1=y-g_i+a\sum^n_{j=1}g_j,$$ where 0 < a<1< na, and I need to prove this: $$\frac{\partial(\sum^n_{i=1}\pi^1_i)}{\partial g_i}=-1+na>0.$$ I am not very ...
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1answer
31 views

Lagrange Multipliers for Implicit Functions

How can I find the minimum / maximum of a function for one variable defined implicitly (f(x, y, z) = c) with a constraint g(x, y) = c on the domain? For example, say you wanted to minimize for z: ...
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15 views

Show running time of algorithm on input of size n is $\Omega$ (f(n))

Basically I'm given this algorithm where I have an array A of integers which outputs an n-by-n array B where B[i,j] contains the sum of the array entries A and asked to give a bound of the form ...
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1answer
810 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 ...
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17 views

Derivation of Efficient Frontier (portfolio optimization) question

In Robert Merton's derivation of the efficient frontier of a portfolio, he minimizes $\frac{1}{2}\sigma^2 $ over the investment weights in each asset, where $\sigma^2$ represents portfolio variance. ...
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2answers
127 views

Jacobi-Method with Projection onto Box Constraint

I'm solving the constrained least squares problem $\underset{u \in [0,1]^N} \min \lVert Au-f \rVert_2^2$ with $u \in \mathbb{R}^N$, $A \in \mathbb{R}^{N \times N}$ and $f \in \mathbb{R}^N$ by using ...
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Is $[u_1,u_2]$ an edge of the polytope $conv(F)$?

Here's the problem. I have a finite set of vectors $F \subset \mathbb{Z}_{\geq 0}^d$. I define $P$ to be the convex polytope $conv(F)$ i.e. the convex hull of $F$. Given $u_1, u_2 \in F$, is the ...
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Designing linear systems to respond to particular kinds of oscillations

Say that I have a linear system which is being perturbed by an oscillating signal of a single frequency, of the form $$ \dot{\vec{x}}(t) = A\vec{x} + B \sin(\alpha t), $$ where $B$ is a vector of 1s ...
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maximizing a function involving factorial.

Can someone suggest a way to calculate the maximum with respect to $x \ge 1$ of: $$f(x)=\frac{1}{x!} \frac{1}{1-c^{1/\binom{x+n-1}{n-1}}}.$$ The constants $c$ and $n$ are parameters such that $c \in ...
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325 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
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confused between convex and non-linear optimziation

I have an optimization function which contains an objective function which contains sum of decision variables, division of sum of decision variables and also product of sum of decision variables. The ...
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18 views

How fast can you split a set of numbers into 2 sets, where the difference of each sum is maximized

How fast can you perform this task? More specifically, if there is a set of 2n elements, how fast could you split those elements into two groups of n elements where the sum of each group is of ...
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For single variable, being $x^*$ a local minimzer, suppose $|x-x^*|=\epsilon$. Find bounds on $|f(x)-f(x^*)|$ and $|f'(x)-f'(x^*)|$.

Im studying for a test on unconstrained optimization and completing exercises from a book that doesn't give the solution to all of them. This is one of them, I aren't sure if I am going the right way: ...
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44 views

Integer Optimization

I have an integer optimization problem that I've been pondering for the last several days. Here's an abbreviated version: I have several wav song files with variable sizes (601201 kilobytes for ...
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Quick help on showing a set is bounded

I am working on a constraint optimization problem. I have found the extrema and all I need to do now is to show that the set S that the critical points are defined in is bounded and closed (therefore ...
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optimisation problem with linear constraint

optimisation problem with linear constraint I have an optimisation problem. I wish to maximise a function subject to a constraint. It is the constraint that is causing me problems. I am using an ...
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Proving inequality equation

Let $a_1, a_2,....,a_n$ be positive numbers such that $\sum_{i=1}^n a_i = 1$ Then for any vector $(x_1,x_2,...x_n) \ge (0,0,...,0)$ I want to show that $$x_1^{a_1}*x_2^{a_2}*...*x_n^{a_n} \le ...
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Equivalence of Two Statements (Duality Theory, Optimization)

Let $a$ and $a_{1}, ... , a_{m}$ be given vectors in $\mathbb{R^{n}}$. Prove that the following two statements are equivalent. $a)$ For all $x \geq 0$ we have $a'x \leq max_{i} a_{i}'x$. $b)$ There ...
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Properties determining boundedness of function

The function I am looking at is $$f(x) = \frac{1}{2}x^TAx + b^Tx + c$$ where $A$ is a symmetric matrix in $\mathbb{R}^{n\times n}$ and $b,c$ belong to $\mathbb{R}^n$ I want to determine what ...
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How to solve the following minimization problem? [duplicate]

For $n$ scalars $a_1,...,a_n$, find the solution of $$\min_{x \in \mathbb R}\sum^n_{i=1} |x - a_i|$$ I denoted $$\delta(x) = \sum^n_{i=1} |x - a_i| $$ and found $$ \delta'(x) = \sum^n_{i=1} ...
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partial derivative of a vector with respect to a variable

I have a vector in the following form $\mathbf{w}^T = [a_1*w_1, a_2*w_2, \dots, a_d*w_d]$ what is the partial derivative of $\mathbf{w}$ with respect to $w_j$ ? (1 or 2) $\frac{\partial ...
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Minimizing 1-dim problem containing quadratic and sum of absolute value functions

I stumbled on a problem which I am not sure how to most efficiently solve - I want a solver in code which I need to repeat several times with various constants. Basically I want to minimize a 1-dim ...
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20 views

How to reduce 3 dimensional optimization to 2 dimensions?

I am trying to minimize the surface area of a parallelepiped of unit volume. Using Volume = xyz(1 + 2cos(a)cos(b)cos(c) - cos^2(a) - cos^2(b) - cos^2(c))^1/2 = 1 where x,y,z are edge lengths and ...
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How to prove that a function is compact (closed and bounded)?

The specific function I am looking at is $f(x_1,x_2) = x_1x_2 + \frac 1{x_1} + \frac 1{x_2}$, where for a fixed $a > 0, f(x) \le a$ and $(x1,x2) > 0 $ I'm really just looking for where to ...
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34 views

The relation between two different definitions of Affine sets

I am following a presentation, which says that for an affine set $L \subseteq \mathbb{R}^n$ it is: $$L=\left\{x|Ax=b \right\}$$ for some $A,b$. The first definition of $L$ as an affine set is given ...
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Separate a list of spheres into several lists, each contained in a sphere with a radius no larger than specified.

I have a list of arbitrary spheres, what I want to end up with is that list separated into a number of groups, where spheres in each group all fit into thier specific larger sphere. The limitation is, ...
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Calculus optimization word problem

"A printed page is being designed to contain up to 96 square inches of printed material. The margins are 1” on the left and right and 1.5” on the top and bottom. Find the outer dimensions of the page ...
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31 views

Fenchel dual vs Lagrange dual

Consider the Fenchel dual and the Lagrangian dual. Are these duals equivalent? In other words, is using one of the these duals (say for solving an optimization), would give the same answer as using ...
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Approximating the max value of a function containing the half-sum of binomial series

I recently met a problem and I have been finding related materials about it. However it is still hard to handle. Precisely, I want to maximize a function related to $$S_n = \sum_{i=0}^{\lfloor ...
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29 views

How to find the parallelepiped of unit volume with minimal surface area?

Is it best to approach this problem using edge lengths and the angles between them? I am trying to reduce the problem to two dimensions, although I haven't successfully done so yet So I have Volume ...