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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right circular cylinder inscribed in a sphere

Find the dimensions of the right-circular cylinder of greatest vloume that can be inscribed in a sphere with a radius of 6 $in$ I think I need help visualizing, and maybe the solution. I've ...
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15 views

Approximating a quadratic term in the constraint set as 2nd order Taylor expansion

I have an optimization problem in the following form: $$\min_{x,y} f(x)+g(y)$$ $$s.t.$$ $$Ax+h(y)=0$$ where $h(y)$ is a quadratic in $y$. Instead of solving this problem directly, $h(y)$ is ...
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28 views

Is there a way to calculate RMS value continuously?

Using that the RMS by definition is: $\sqrt {\int_0^T\frac 1T*f(t)^2dt} $ which can be calculated by using Riemann sums in the following way: $\sqrt {\frac 1N\sum_0^Nf[i]^2} $ I've tried that in ...
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1answer
50 views

Derivative of log-likelihood cost function with respect to a matrix

Recently, I am learning derivative method to a function and thanks to @hans help, I can solve those which can be expressed by Frobenius product. But for the log-likelihood function, I do not how to ...
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0answers
21 views

Maximization of quadratic form on a sphere [duplicate]

I have to following problem $$\max_{x}x^TAx+b^Tx\quad \mathrm{s.t.}\quad x^Tx\leq c,$$ where $A$ is real, symmetric and positive semi-definite. Firstly I tried to solve the problem with the KKT, but ...
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6 views

transformation of one maximization problem

I want to maximize a function $h(x,y)=f(x,y)-g(x,y)$, subject to (1) $0 \leq g(x,y)\leq I$, where $I$ is a fixed positive number; (2) $x\geq 0$ and $y \geq 0$. I come up a method to solve this ...
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1answer
21 views

What is the class of this Integer programming prob.

I have an optimization problem which seems to be non-linear because of the constraints (right?): $max (\sum U_i\times x_i)\\ \sum x_i\times y_i\times r_i\leq R\\ \sum y_i=1\\ \sum x_i=1\\ x_i, ...
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0answers
14 views

Analytic solution to semidefinite programming

Problem \begin{align} &\arg\,\min\limits_{0 \le \rho \le 1} \rho \\ s.t.& \begin{bmatrix} A P A - \rho^2P & A^TPB \\ B^TPA & B^TPB \end{bmatrix} + \lambda \begin{bmatrix} C & ...
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0answers
35 views

Increase max-flow by 1 with minimum changes to edges

Suppose we have a directed graph and we have the maximum flow from $s$ to $t$ as $f$. Now we want the graph to have a flow of $f+1$. This requires us to increase the capacity of a certain subset of ...
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0answers
21 views

Coordinate descent with equality and inequality constraints

I have an intuitive understanding of why the simple method of coordinate descent does not work with linearly coupled constraints such as; $$\min_x\sum_if_i(x_i)$$ $$s.t.$$ $$Ax=b$$ If we try to ...
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1answer
24 views

Local extrema of $x^3+y^2+6y$

I have to find local extrema of $x^3+y^2+6y$. I found out that the stationary points are $(0,-3)$. I also found the Hess matrix for this function and computed the determinant, which is $12x$. But now ...
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0answers
19 views

Minimizing the average

Let's say I have a nice-behaving function $f: \Bbb R^n\to \Bbb R$, and I would like to find its maximum. Then I can apply gradient search algorithms to look for that, and to cope with possible ...
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0answers
11 views

Rank one correction algorithm

Are the direction d1,d2,...,dn necessarily conjugate? enter image description here
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23 views

How to numerically minimize system of equations composed of data and smoothness terms, ensuring minimum solution norm

I need to find $g$ that minimizes: $$\sum_{v=0}^n (f+g_{v_{left}}-g_{v_{right}})^2 + \frac{1}{\lambda}\sum_{v=0}^m (g_{v_i}-g_{v_j})^2$$ where $f$ is constant and the sums are over pair of $v$ ...
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1answer
23 views

Sum greater than 1; minimization for non-strict inequalities

I want to show that if $x_k>0$ for $k=1,2,...,n$ and $\sum_{k=1}^n x_k=1$, then $\sum \frac{x_k^2}{y_k}\ge 1$ for any $y_1, y_2,...,y_n>0$ so that $\sum_{k=1}^n y_k=1$. I tried solving the ...
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0answers
8 views

Closed-form solution for system of equations for finding a critical point

I am trying to find a critical point of a function $\mathbb{R}^d \to \mathbb{R}$ by setting its gradient to zero. I would like to solve the follwoing system of equations. $$\frac{1}{1 - \sum_{j=1}^d ...
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0answers
24 views

Maxima/Minima question seems contradictory

Sorry for putting in the picture.I tried but I wasn't able to input the inverse function using Latex. So my question is as given in no. 21. It states that, the function is minimum at $\ x=1$.This ...
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0answers
25 views

Optimization problem regarding Newton's algorithm

I would want to ask why does Newton's algorithm with Wolfe line search converges to (0,0) no matter where the starting point is?
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21 views

Compute smoothed probabilities for EM algorithm

In order to compute the expected value of log-likelihood in EM algorithm, we use 3 different probabilities Forecast (predictive) probabilities Inference probabilities Smoothed probabilities ...
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1answer
44 views

Solving optimization with Lagrange multipliers

I am fairly new to Lagrange multipliers. Can someone please show me how to maximize the following function: \begin{align} f(x,y)=240\sqrt{x}+y \end{align} Subject to: \begin{align} 30x+y=720 ...
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1answer
50 views

calculate price based on demands and maximize revenue

I believe I have a simple question which I am struggling to answer. It is as follows: We have 400 items, each item costs £100. Retailer bought these items before the season started. The forecasted ...
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1answer
46 views

Linear Programming: Maximize

Jimbo Enterprises produces $n$ products. Each product can be produced in one of $m$ machines. Let $t_{ij}$ be the time in hours needed to produce one unit of product $i$ on machine $j$. For month $k$, ...
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1answer
25 views

Subgradients of non-convex functions

In these notes (section 2.3), it is stated that: A point $x^*$ is a minimizer of a function $f$ (not necessarily convex) if and only if $f$ is subdifferentiable at $x^*$ and $0 \in\partial ...
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1answer
59 views

Solving a linear program thanks to complementary slackness theorem

Using the complementary slackness theorem, say if the following basis optimal: $$x_1*=0=x_5*,x_2*=4/3,x_3*=2/3,x_4*=5/3$$ \begin{cases} \max & 7x_1 ...
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30 views

How do I get the pattern for highest profit (by rate of increase per variable) out of a complex equation?

WARNING: Complicated as hell. If you are a League of Legends player then you might understand more. The formula I'm looking at is $AD*AS*(1+ASB)*[(1-CC)+CC*CF]$ I'm trying to figure out which ...
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1answer
51 views

Assuming $0 \leq a_{n+1} \leq c_n a_n + b_n$ (+ other conditions), show $a_n \to 0$

In the paper "A primal-dual splitting method for convex optimization ..." (see here https://www.gipsa-lab.grenoble-inp.fr/~laurent.condat/publis/Condat-optim-JOTA-2013.pdf), Lemma 4.6 states the ...
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1answer
53 views

Max $z = x_1(1-x_2)x_3$ s.t. $x_1 - x_2 + x_3 \le 1$

Using dynamic programming, Maximise $$z = x_1(1-x_2)x_3$$ subject to $$x_1 - x_2 + x_3 \le 1$$ $$x_1, x_2, x_3 \ge 0$$ Here's the outline of my solution 1. How is it? Let ...
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Second Order Conditions and Maximum Likelihood Estimator for Normal and Exponential distributions

I can't seem to show the Second Order Condition for the MLE of the exponential distribution is <0. Does anyone have any hints? Same problem for the normal distribution when looking for MLE of the ...
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1answer
38 views

How to derive 2D equation representing minimums of constrained 3d equation?

I have a 3D (multivariate) function f(x,y) which can be represented as a surface with constraints. When the surface is viewed from the side (as below), such that the Y axis is not visible, there is ...
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26 views

Image processing, optimization via regularization - efficient strategy

i would like to solve the following system: $J(x) = |Ax-b|_2^2+\gamma|\nabla x|_2^2$ subject to: $x \geq 0, \sum_i x_i = 1$ The underlying problem is to derive the PSF from a sharp and blurry ...
2
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1answer
29 views

Use of binary variables in LP problems

I can't figure out how to write the following condition to an LP. I have four nonnegative variables: $X_A$, $X_B$, $X_C$, and $X_D$. The condition which should be satisfied is this: If $X_A$ and ...
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1answer
29 views

Help with minimizing integral

I want to make a formal argument that for the following optimzation problem $\underset{S}{\operatorname{argmin}} \int_0^D (x(t) - S)^2$ the minimum solution is to set S to the mean of x(t) in the ...
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0answers
22 views

Proof of orthogonality in the gradient descend algorithm.

Ok, this is perhaps an easy question but I'm stuck, so any help will be cherished. The gradient descent algorithm updates the weights as: $$\textbf{w}_{t+1} = \textbf{w}_{t} - \eta\nabla ...
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25 views

Limit of complex function at infinite order

Hello :) I'm trying to prove a theorem which is showing to be little difficult to do...So the problem is to prove the following: \begin{equation} \lim_{n\to\infty} \sup_\omega \left\lvert ...
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Minimizing the distance between two set of vectors such that the angle of both set is equal

Suppose I have two set of vectors K1,I1 and K2,I2 forming a surface S1 and S2 respectively in R2 or R3. The angle between K1 and I1 is T1 and K2 and I2 is T2 respectively. The goal is to minimize the ...
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0answers
17 views

Optimization of area of rectangle within semicircle [duplicate]

The semi-circle is given by $y=\sqrt{25-x^2}$ Find the length and width of the rectangle such that it's area is optimized. How do I deal with problems such as these?
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9 views

Optimization Problem, including block bids.

Block order optimization Hello, We're a bit stuck on this problem, which involves bidding in blocks. We're given $Q, K, s(1),s(2),...s(24)$ $$ \underset{q}{\text{maximize}} ...
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5 views

Does one counterexample suffice to show that LPT-rule is not optimal for $P \mid \mid C_{\max}$ when $\#\text{jobs} \leq 2\cdot \#\text{machines}$

Excuse me for a somewhat trivial question, but my I can't seem to find closure. For a homework assignment, we are asked to show the following: Consider the problem $P \mid \mid C_{\max}$. ...
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6 views

Given M points and a weighted graph G, map the vertices to distinct points to minimize sum(edge_weight*edge_length)

Given an arbitrary undirected weighted graph G with N vertices, and an arbitrary set of M points P in euclidean 3-space, where M>=N, map the vertices to distinct points such that sum(edge_weight * ...
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16 views

Preservation of monotonicity under argmax

Suppose $f(x,y)$ is non-increasing in $y$ for all $x \in X$. Then, can we show that $x^*(y) = argmax_{x \in X}\{f(x,y)\}$ is also non-increasing in $y$? If so, what characteristics should the function ...
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Impact of removing active constraints in convex optimization

In active set methods for non negative least squares, we remove variables from the passive set to active set if the least squares solution gives negative values on those variables. What's the impact ...
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1answer
14 views

Chebyshev's approximation understanding

I am reading Boyd's book on convex optimization. Could you assisst me in understanding what this expression means: $$\text{minimize} \ \ \text{max}_{i=1,...,k}|a_i^Tx-b_i|$$ This is what I think ...
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28 views

Nonlinear optimization with eigenvalue problem as a constraint

I have an unknown matrix $\mathbf{A} \in \mathbb{R}^{2n \times 2n}$ which is a function of $n$ parameters $a_i, i=1,2,...,n$. The objective is to find these $a_i$'s and the objective function is as ...
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1answer
34 views

Lagrangian Multiplier for liner problem

I have a (probably) stupid question but I can't find the answer. I have the following problem (my problem is much more complicated but as an example) : \begin{equation} \begin{matrix} \displaystyle ...
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2answers
34 views

Computing operator norm of a matrix

In my notes I have that $\left\|\, \begin{bmatrix}3&1\\1&1\end{bmatrix}\,\right\| = 2+\sqrt{2}$; but I'm struggling to get this. Here's what I have: $$\left\|\, ...
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0answers
19 views

Semidefinite programming with symmetric matrix constaints

\begin{align} &\arg\min\limits_{0 \le \rho < 1} \rho \\[1ex] s.t.\quad & \begin{bmatrix} 1 - \rho^2 & -\alpha \\ -\alpha & \alpha^2 \end{bmatrix} + \lambda \begin{bmatrix} -2mL ...
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0answers
26 views

If the primal is unbounded, then the dual is infeasible.

In the context of duality in linear programming, prove that If the primal is unbounded, then the dual is infeasible. My book says that this is a corollary to complementary slackness. What's ...
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1answer
17 views

LAD analytical minimization

Is it possible to minimize least absolute deviations analytically? Say given a sample $\{x_i\}_{i=1..n}$ find $$\arg\min_\lambda{\sum_{i=1}^{n}{|x_i-\lambda|}}$$
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1answer
20 views

Can a network migration problem be solved with linear programming

I'm trying to solve, using linear programming, the problem of determining in which order should network elements by migrated from one place to another. The idea is that resources such as bandwidth ...
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64 views

Heuristics for topological sort

I have a number of modules connected in a Directed Acyclic Graph. My problem is to find an optimal execution order (minimize the total execution time). Any topological sort suffices for a valid ...