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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$f(z)=\frac{1}{z^2-2z+2}$ - Maximum modulus principle

Let the function $f(z)=\frac{1}{z^2-2z+2}$. I have to find $\max_{z \in D(0,1)} |f(z)|$, but I already know that the maxixum would be on $\bar{D}-interior(D)$ by the maximum modulus principle. Is ...
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On Boyd et al.'s convergence analysis of ADMM: Why do we need the convexity assumption?

Please refer to Boyd et al.'s convergence analysis of ADMM (Chapter 3 and Appendix A). My question is: Why do we need $f$ and $g$ to be convex? I don't see the need of this assumption. If the ...
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Euler-Lagrange Extremal Value Functions

In this question, part a is self-explanatory. However, for parts b and c ( F = y' and F = y^2) how would this work? Part b E-L gives d/dx (1) = 0, which is correct, but then which function y ...
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Converting generic linear problems into their dual

I'm revising how to do dual problems in linear algebra. I'm very weak in Linear programing but I struggle to cope with the topic during lectures and assignements. I have to convert the following ...
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30 views

Partition a triangle into equal areas

A piece of wooden board in the shape of an isosceles right triangle, with sides $1$,$1$, $\sqrt{2}$ is to be sawn into two pieces. Find the length and location of the shortest straight cut which ...
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How to deal with an $xy\le 1$ constraint?

I have to solve the following optimization problem: $$ \begin{align*} \min_{x,y} &\{-x-y\} \\ \text{such that} \\ y &\ge 3 \\ y &\le 30 \\ x &\ge 0 \\ xy &\le 1 \\ \end{align*} $$ ...
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1answer
16 views

Dealing with non-negativity constraints without using Kuhn-Tucker conditions

Suppose I wish to maximize the function $f(x,y)$ subject to the equality constraint $g(x,y)=c$ as well as the non-negativity constraints $x\geq0$, $y\geq0$. If I first solve it ignoring the ...
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looking for polynomial with degenerate local (not gloabl) minimum [closed]

Can anyone give me a multivariate polynomial $f$ such that the origin is a degenerate critical point and also a local (not global) minimizer of $f$? Thanks in advance!
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4answers
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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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14 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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27 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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40 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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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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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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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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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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26 views

Convex optimization qualifying exam [closed]

I'm studying for my qualifying exam which I'm going to take in late July and I have some problems from previous exams that I could't solve . So I would appreciate your help because I'm so stressed ...
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34 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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18 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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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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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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Rank one correction algorithm

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

Sequential convex second order cone programming [closed]

I am trying to solve an optimization problem where the objective function is convex, the inequalities are second order cones but I also have nonlinear equality constraints. The convex energy can be ...
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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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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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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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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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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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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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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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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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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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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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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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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
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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 as illustrated here. When the surface is viewed from the side as shown here, such that the Y axis is ...
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25 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 ...
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1answer
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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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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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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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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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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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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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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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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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Maximize the magnitude of a complex function

Given $$ B(\phi) ~=~ \cos(\phi - \phi_L) + \cos(\phi - \phi_L - \tilde{\phi}) ~e^{j{2\pi}d \left[(\cos\phi - \cos\phi_L )\;+\;(\sin\phi - \sin\phi_L) \right]}, $$ $d > 0$. $\phi_{m} = \arg ...