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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Optimizing a problem using Lagrange multipliers

$\newcommand{\norm}[1]{\|#1\|}$ I have the following problem: $$ \min_{w,\theta}\frac{1}{2}\norm{w-w_t}^2+\frac{1}{2}(\theta-\theta_t)^2 \text{ s.t. } w^\top(z(n-\theta)-\hat z(\hat n - \theta)) \ge 1 ...
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AI Parameters for Tetris-like Game

I am building an AI to play a variation of Tetris. The rules are changed in that there are 19 different types of pieces, rotation is not allowed, and the pieces can be placed anywhere in a 10X10 grid. ...
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Formulating deterministic and stochastic production models (not solving them) [Beginner's Operations Class]

Question provided in picture This question has been troubling me as I am not used to questions without numbers as it is hard for me to visualise. I also find stochastic problems hard in general. &...
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General process to find global extrema of a function?

I have been reading and watching videos about local and global extrema, but all of this material covers the topic just graphically, and nobody really explicitly cares on how to find the global maximum ...
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Finding minmax over set of constrained continuous functions

I have this problem that I've been struggling with for a bit, and I don't know how to proceed. I'm looking for the solution to: $$ \min_f \max_s \frac{\int_s^t f(x) \;dx}{\int_s^\infty f(x)\; dx} $$ ...
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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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Formulating an optimisation problem into a mixed-integer problem

I'm not sure if I understand this question and was wondering if anyone could provide any insight to an answer. The only thing I can think of adding is a constraint: "x2 = integer", so I'm clearly ...
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Solving Optimization Problem (Orthogonal Projection) Using Projected Sub Gradient / Dual Projected Subgradient

Given the following optimization problem (Orthogonal Projection): $$ {\mathcal{P}}_{\mathcal{T}} \left( x \right) = \arg \min _{y \in \mathcal{T} } \left\{ \frac{1}{2} {\left\| x - y \right\|}^{2} \...
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LP: add extra costs in the objective function for every variable which is not equal to $0$

I am trying to optimise an LP problem but extra costs should be added if a variable is larger than $0$. For example, if we have the following objective function: $$\text{minimize} \qquad 2X_1 + 3X_2 ...
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Is generalized mean convex / concave?

The generalized mean can be given using the following equation: $ M_p(x_1, \dots, x_n) = (\frac{1}{n}\sum_i x_i^p)^{1/p} $ Is it convex /concave when $p<1$ ?
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Campbell's Source coding

In the usual Shannon's source coding problem one chooses code words that minimize $E[L]:=\sum_i p_il_i$ over all $L=(l_1,l_2, \dots), l_i\ge 0$ such that $\sum_i e^{-l_i}\le 1$ (Kraft inequality), ...
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Conjugate of difference of convex functions

I am reading through this tutorial on DC programming and the author makes a startling claim without proof: If $g$ and $h$ are two lower semi-continuous convex function, then the conjugate function of ...
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1answer
609 views

Maximize profit with dynamic programming

I have 3 tables… $$\begin{array}{rrr} \text{quantity} & \text{expense} & \text{profit}\\ \hline 0 & 0 & 0 \\ 1 & 100 & 200 \\ 2 & 200 & 450 \\ 3 & 300 & 700 \\...
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Interplanetary Optimisation using a simulator with PyGMO or SciPy

I am currently trying to use a N-body gravity simulator to model a spacecraft trajectory and using the simulator as a BlackBox to optimise the trajectory. I am thinking of using basin hopping/ ...
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25 views

Connection between complementarity problem and optimization problem?

I do not understand the connection between complementarity problems and optimization problems. I have tried to look at other definitions for complementarity problem to see if that would help me with ...
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401 views

Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
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Variational Inequalities - What excatly does the definition say? Why are they useful?

I am having issues understanding the definition of variational inequalities. We have the following definition: Given a set $X \subset \mathcal{R}^n$ and a mapping $F: X \rightarrow \mathcal{R}^n$ a ...
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Minimize two-variable function

I wish to minimize a function of two variables $m$ and $L$ (both strictly positive). I have calculated the first two partial derivatives: $$\left[ \frac{-n}{2L} + \sum_{i=1}^n \frac{ (x_i - m)^2}{2m^2 ...
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470 views

Using Lagrange multipliers to find the shortest distance between two straight lines

A problem asks me to use the method of Lagrange multipliers to find the shortest distance between the straight lines $x=y = z$ and $x = -y, z=2$ (It also warns me that using this method is a bit ...
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Calculate best interval between peaks

I have a vector of values with zeros and some rare positive value (corresponding to the peaks in the hist) ...
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Newton conjugate gradient algorithm

In this video, the professor describes an algorithm that can be used to find the minimum value of the cost function for linear regression. Here, the cost function is $f$, the gradient is $g_k$ where $...
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optimization equivalence

Given the functions $f_1(r,x)$ and $f_2(r,y)$: $[0,1]\times \Bbb R \to \Bbb R ^+$, solve the following problem $$\underset{r,x,y}{\text{argmin}}\; f_1(r,x)+f_2(r,y) \\ \text{subject to}\; x^2+y^...
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How can I express the minimization of the p90th percentile mathematically?

I would like to minimize the 90th percentile of a function with a normally distributed variable. If I wanted to minimize the expected value, I would do it something like this: $$ min_s \ z = E(f(X,s)...
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Deducing MaxFlowMinCut from Menger

So the MaxFlowMinCut theorem with rational network capacities and (the edge-version of) Menger's theorem for undirected graphs are equivalent, both directions being not too hard. I gather that since ...
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How to programatically solve the optimal control problem?

I have to programatically (write a program) find a control function $u(\cdot)$ to minimize the following functional: $$ J(u,x) = \int_0^T { f_0(x(t), u(t), t)}dt + \Phi(x(0)) \rightarrow \min$$ ...
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How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
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How to find the value of lambda in following optimization problem?

Consider following optimization problem: $P_T = $minimize$(p_s + p_r)$ Subject to $p_s \ge p_{s,min}$ and $p_r \ge 0$. After solving the optimization problem, we get following equations for $p_s$ ...
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Enumerating (some) combinations of elements subject to a constraint

Consider this variant of the knapsack problem: I own an outdoor goods store, and hikers come from miles around because of my amazing variety of products for sale. There are 4 popular hikes in the ...
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Extending the ordered sequence of 'three-number means' beyond AM, GM and HM

I want to create an ordered sequence of various 'three-number means' with as many different elements in it as possible. So far I've got $12$ ($8$ unusual ones are highlighted): $$\sqrt{\frac{x^2+y^2+...
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Trigonometry minimum value

If $\alpha , \beta$ and $\gamma$ are angles of triangle How can we prove that $$ \cot^2(\alpha) + \cot^2(\beta) + \cot^2(\gamma) $$ has a minimum value of $1$. I actually used the AM-GM inequality ...
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How can I solve this optimization problem?

How do I solve this optimisation problem? $$W = \left(\frac{n(X-Y-Z)p}{Zq}\right)^{1/a},\, a>0$$ $\operatorname{Max}\{ W\}$, subject to $0\leq n \leq 1$, $0\leq Y \leq X$ and $Z \leq Z_{max}$ ...
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How to solve the following optimization problem with projection?

How to solve the following optimization problem with projection? \begin{alignat}{1} &\min_{u_+,u_-,s,l\geq 0} \frac{1}{\lambda} \langle A ,(a +u_+-u_-)(a +u_+-u_-)^\mathsf{T} \rangle+\mathbf{1}^\...
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Is this the correct way of using Variational Principle (Minimization Principle)?

I am constructing a smooth function $f(x)\equiv f(u(x),v(x))$, such that $u(x)$ and $v(x)$ are some trial parameters. I have the following integral $$G=\int_{x_i}^{x_f} f(u(x),v(x)) \mathrm{d}x.$$ My ...
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Minimizing the Frobenius norm with linear inequality constraints

How to solve the following system for $\mathbf{C}$ and $\mathbf{a}$: $\min\|\mathbf{X-XC} \,\mbox{diag} (\mathbf{a})\|_F^2$ subject to $\mathbf{c}_{ik}\geq 0$, $1^T \mathbf{c}_k = 1$ and $1-\delta\...
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The Dual problem of a non constraints problem?

The primal problem is $min_{w\in R^d}: P(w)$ where $P(w)=\frac{1}{n}\sum_{i=1}^n\phi_i(w^Tx_i)+\frac{\lambda}{2}||w||^2$. The dual problem is $max_{\alpha\in R^n}: D(\alpha)$ where $D(\alpha)=\frac{...
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LP problem: Giving variables the same value or 0

If I have the following objective function: $$\min X_1 + X_2 + X_3 + X_4$$ How could I ensure that the variables $X_1, X_2, X_3$ and $X_4$ either have the value of 0 or they could have a random ...
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How to minimize a linear function over a halfspace efficiently and intuitively

Consider the following fundamental problem: Two methods: By duality: ($\lambda, b \in R$) $L(x,\lambda)=c^Tx+\lambda(a^Tx-b)=x^T(c+\lambda a)-\lambda b \ \ $. Therefore, $g(\lambda)=-\...
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Equation of the form $\mathbf{\Phi}'(t)=\mathbf A(t)\mathbf{\Phi}(t)$.

Let $\mathbf{\Phi}(t)$ and $\mathbf A(t)$ be matrices satisfying the differential equation $$ \mathbf{\Phi}'(t)=\mathbf A(t)\mathbf{\Phi}(t)\ . $$ If I am not mistaken, if $\mathbf A$ and its integral ...
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Which way should you run from the lions?

This is a fun problem that I saw somewhere on the internet a long time ago: Suppose you are at the center of an equilateral triangle with side length $s$. At each of its vertices, there is a lion ...
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1answer
42 views

Is Lagrangian Multiplier Equivalent to Brute Force for binary decision variables

I have a set of variables $x_{i} \in \{1,k\} $ in a non linear optimization problem. As this variable has only two possibilities I have encoded this into a constraint. I assumed having equality ...
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Finding a good difficult example function to minimize

I am comparing some code for non-linear function minimization in multiple variables, like quasi-Newton methods etc. I am looking for a nice function to use as a test case. So far I have been using $f(...
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Minimal lateral surface of a cylinder

Inscribe in a given sphere a cylinder such that its lateral surface (without the bases) shall be maximal. So lateral surface is = $2 \pi rh $ $ \Rightarrow 4 \pi x \sqrt{r^2-x^2}$ Now take ...
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Why do we minimize the squared norm instead of the norm in this optimization problem?

When reading about the optimization problem for Support Vector Machine in Bishop's book (Pattern Recognition and Machine Learning) he wrote that: The optimization problem then simply requires that ...
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Finding equation of a bent sufficiently flexible cardboard of length $l$ fitting into a gap of width $m<l$

I was thinking about how the walls of a barrel is made then I realized it is someone like fitting a piece of wood of length $l$ in between some "gap" of length $m<l$. This would cause the piece of ...
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Question on optimization algorithm to train peculiar regression

I've been in my operations research course, and we have been working on optimization in particular problems within regression. We hypothesize that for variables $h,s,d,t,$ there is this set ...
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439 views

Minimize Frobenius norm with constraints

As a follow-up on my previous question, I would like to solve the following optimization problem: $\min \Vert MA-B \Vert_F^2-x^HMy\;\;s.t.\;\;M^HM=I$ where $A$ and $B$ are $N\times L$ complex ...
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Finding the minimum of $x_1 + \cdots + x_n$ on ellipsoid

Let $A$ be a positive definite matrix $n \times n$ and $u^T = [1 \cdots 1]$. Use Lagrange multipliers to find the minimum of $f(x) = u^Tx$ on $h(x) = \frac{x^TAx}{2} = 2$ This is what I did. $$L(x,...
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Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
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Can $n$ variables ever have more than $n$ unique satisfiable constraints?

Assuming you have $n$ variables, how many maximum independent satisfiable constraints can you have? What I mean by independent is that the equations all express unique constraints, s.t for example $x +...
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Sort a set of points to minimize the sum of the square distances between two consecutive points

Let $P$ be a finite set of points in $\mathbb{R}^3$. Let the number of points in $P$ be $n\in\mathbb{N}$. I want to sort the points in $P$ to minimize the sum of the distances between two consecutive ...