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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Max-Min values of $f(x,y) = x^3+y^3-6x^2-y-1$

I am asked to find the extrema of the function $$f(x,y) = x^3+y^3-6x^2-y-1$$ I understand that we have to equal the partial derivatives to zero, which means $$ f_x = 3x^2-12x = 0\\ f_y = 3y^2-1 = ...
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Issue with optimization problem

Consider the following configuration: Now, we can minimize the length $L=2\ell_1+\ell_2$. Let the top left angle be $\theta$ so that $\ell_1=\frac{a}{2}\sec\theta$ and ...
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Inverse Vectorization Vec^-1

Hope that you will find this post in good health. I am Mr.Adnan from Pakistan with research background in Control systems. I am working on one problem in which Hadamard weights are using. During ...
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28 views

Integer programming with linear constraint

I am trying to find the optimal solution for the following problem \begin{eqnarray} &&\underset{x_i, ~y_i ~\forall i} {\text{maximize}} ~ \sum_{i=1}^N x_i f_i(y_i) \\ && ...
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Local minimum for a polynomial

Can someone please help me to answer this question: we consider $V \in E_r =\lbrace P \in \mathbb{R} [X_1,X_2,..,X_d] \mid \deg P \le r \rbrace$. If $\exp^{-V(x)}\in L^2(\mathbb{R}^d)$ then $V$ ...
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How to solve a linear program with additional equality constraints?

The following optimization problem $$\max_{\substack{x \ge 0,\\Ax^T+b^T\ge 0}} c x^T$$ where $x \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, $b\in\mathbb{R}^m$, and $c \in \mathbb{R}^n$ is ...
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Distributed control problem which involves the p-Laplacian operator

Someone could help me to deduce the optimality system for the optimal control problem: \begin{align} &\min_{u\in L^{2}(\Omega)} ...
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47 views

Questions about simplex algorithm

I'm trying to understand how simplex algorithm works, and here are my questions: 1. Why we choose the entering variable as that with the most negative entry in the last row? My understanding is that ...
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15 views

Minimum number of $m \times m$ matrices needed to recover a single large matrix

This problem was motivated by the need to efficiently train a neural net on a dataset in which the labels represent dependencies between examples, but nothing about it is machine-learning specific so ...
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Different ways of solving $\underset{\mathbf{s}}{\text{min}}\;\|F\mathbf{s}-\mathbf{x}\|_{l_2}^2 + \|W\mathbf{s}\|_{l_2}^2$ least square problem?

The problem that I am going to describe arises from compressed sensing technique and after using weighted least squares it can be transformed into the following least squares problem: ...
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46 views

Values of parameter $\epsilon \in (0,1)$ that make a rational function decreasing

For $p \in (0,1]$, an integer $n \geq 2$ and $\epsilon \in (0,1)$, I want to show that $$\frac{p (1- \epsilon p)^{n-1}}{1- (1-p)^n}$$ is a decreasing function of $p$ for $\epsilon > g(n)$ for some ...
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how I can minimize this equation using derivation

I'm a software engineer and have not much mathematical knowledge. Now, I'm facing with a problem in my research. I have a system of equations as below: $$P_1 = \alpha V_p + \beta I_c^2 $$ $$P_2 = ...
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24 views

Drift management optimization

I have a problem in which I am having trouble formulating the optimization. A portfolio value is $10M I have a vector of current weights [.10,.15,.15,.10,.05,.10,.20,.15] and another vector of ...
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10 views

Maximize function symbolically

I have the following expression: $$ \sum_{i,j=1}^n\rho_{ij}^2-\frac{2}{n}\sum_{i=1}^n\left(\sum_{j=1}^n\rho_{ij}\right)^2 +\frac{1}{n^2}\left(\sum_{i,j=1}^n\rho_{ij}\right)^2 $$ My goal is to ...
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31 views

Optimization Cost of candy

You have decided to buy candy for the trick-or-treaters and have estimated there will be 200 children coming to your door, and plan to give each children three pieces of candy. You have decided to ...
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generalised eigenvalue problem with absolute value

Problem: $\max_w |w^t A w|-|w^t B w|$ s.t $w^t C w=1$ If there was no absolute values, i.e. if the problem was $\max_w w^t A w-w^t B w$ s.t $w^t C w=1$ this would, by using the appropriate Lagrange ...
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Bipartite Matching with quadratic objective

I'm looking for the best way to formulate and solve the following bipartite matching problem: I have n nodes on the left hand side of the diagram, partitioned into ...
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21 views

Why can't I use sum of probabilities as my loss function for machine learning?

I'd like to understand what is the major reason that we are using loss function of the following form in machine learning (I know it is obtained by taking a logarithm of the likelihood of the ...
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2answers
30 views

Optimisation of a juice box: finding the least possible surface area that can hold the most volume

I have an investigation which requires me to design the dimensions of a juice box (cuboid) which has the least possible surface area that can hold the most volume. I am not sure as to how I should ...
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7 views

Can an infeasible point be used to initialize an Active Set Method (optimization)

Consider an optimization problem with a quadratic objective function and linear inequality and equality constraints. Consider an Active Set Method for optimization. Say you do not know a feasible ...
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How do I classify extrema found using Lagrange multipliers?

Ok so I have found a bunch of local extrema using the method of Lagrange multipliers. Now how do I classify them as minimum or maximum? I cant use the second derivative test because its not a ...
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6 views

Going from discrete solution to continuous solution (Dynamic Programming/Optimal Control)?

Suppose I have a discrete solution for a dynamic programming problem and an optimal control policy. If I can make the control policy continuous, by taking the limit as t-> 0, is that control policy ...
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Disadvantages of particle swarm optimization method

I am using particle swarm optimization method. It has a lot of advantages, but I am looking for disadvantages of this method. Can you help me?
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Exact and Heuristic Optimization Methods

Could anyone give me a rough classification for which kind of nonlinear- problems can I apply exact optimization methods (such as barrier function) and for which problems heuristic methods (such as ...
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21 views

minimize the perimeter

Consider a window the shape of which is a rectangle of height $h$ surmounted by a triangle having a height $T$ that is $0.5$ times the width $w$ of the rectangle (as shown in the figure below). If ...
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20 views

Choosing the knots for a linear interpolation

I want to approximate a function through piecewise linear interpolation and try to understand how I could set the associated interval points optimally. Take a continuous function $f: X \rightarrow ...
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26 views

Determining the most appropriate set of eigenmodes for a modal decomposition of an experimental data set

I have a complex vector of the transverse amplitude and phase distribution of a laser beam, derived from experimental data. When modelling these field distributions, ordinarily the eigenmodes of the ...
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Model linearly: What products to make, how much to make and in what plants to make them?

A company wants to make 3 new products for the upcoming week. We are given that: Each product can be made in 1 of 2 plants. At most 2 of the 3 new products should be chosen to be made. Only 1 of ...
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16 views

Minimizing sum of minimum

What are some theory/algorithm that talk about minimizing sums of minimums? For example, assuming y and z are discrete and the function is linear in x: $\min_{x} \sum_{y} \min_{z} f(x,y,z)$. I ...
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Calculus optimization problem leads to a quartic polynomial - is there a better way?

I am tutoring a student in first-semester Calculus. He needs to minimize the function $$f(x)=\frac{\sqrt{4+x^2}}{2}+\frac{\sqrt{1+(3-x)^2}}{4}$$ Taking the derivative and setting it equal to zero, we ...
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Normalizing eigenvectors when diagonalizing

Suppose your square matrix is symmetric and I want to diagonalize it. Why is it that at the end you normalize the eigenvectors to get your orthogonal matrix (actually orthonormal matrix)? Is this just ...
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Model linearly: Determine amount of units for production

A company produces 2 products in a week. Let $x_i$ denote the number of units of product $i$ to produce. Each product requires liters of Chemical X to make. Info is given below: \begin{array}{|c|c|} ...
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Maximize sum of squares

Lets say that I know that $n$ values $x_i$ sums up to $\mu$: $$ \mu=\sum_{i=1}^n x_i $$ I also now that $0\leq x_i\leq 1$ for all $i=1\cdots n$. I want to find an upper bound as tight as possible ...
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Relationship of radius of sphere to an inscribed right circular cylinder for max and min values

I cannot seem to find the correlation between having an interval of a radius of a sphere with finding the greatest lateral surface area of a right circular cylinder inscribed in it. The question goes ...
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Optimization PDE

I have an assignment where the question reads: $\min J(u) = 1/2 \int_0^1x^2u'(x)^2 dx - \int_0^1 u(x) dx$ with $u \in H_0^1(0,1)$ show $J(u) \geq -1/2 \forall u \in H_0^1$ So I try the usual ...
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$\max 2x_1 +x_2$ unbounded or unfeasible with the constraint $sx_1 +tx_2\le-1$

\begin{cases} \max & 2x_1 &{}+x_2\\ & sx_1 &{}+tx_2&\le-1\\ & x_1,x_2&&\ge 0 \end{cases} Find out when this program is not feasible, bounded Feasibility It ...
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26 views

Show convexity of $f(x,y,z)= x^2+y^2+z^2+xyz$

Let $f(x,y,z)= x^2+y^2+z^2+xyz$. Show that $f$ is convex on $\Omega=${$(x,y,z)\in R^3 : x^2+y^2+z^2<\frac{5}{2}$}. To prove it, I want to show that $\nabla^2f(x,y,z)$ is positive definite. I ...
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Does the existence of a Algebraic Riccati Equation implies the existence of an functional minimization?

Let $\forall k\ge 0. V_k(x)$ be the value function related to the recursive optimization problem $ J(x_0) = \underset{u}{\inf} \sum_{k=0}^{N-1} x_k^T Q x_k + u_k^T R u_k + x_{N}^T P_N x_N \\ s.t. ...
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40 views

How to rewrite/solve this differential equation

\begin{equation} \sin(\theta + d\theta) = \sqrt{1 + \frac{dy}{y}}\cdot{\sin(\theta)} \end{equation} I think this is a non-linear and non homogeneous first order equation. I found this whilst trying to ...
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2answers
20 views

A differentiable function $f$ with maximum at $x=c\Rightarrow f''(c)<0$ true OR false

State true or false A differentiable function $f$ with maximum at $x=c\Rightarrow f''(c)<0$ I think this is a true statement but my book says this is a false statement.I do not understand why ...
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37 views

How many stationary points for this class of functions?

Let $f,g \in C^{\infty}[a,b]$ such that $f(a) = g(a)$ and $f(b) = g(b)$ and $f',g' \leq 0$ and $f'' > 0$ and $g'' < 0$. By the Rolle theorem I can say that there's at least one stationary point ...
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32 views

Weights in goal programming

I'm not quite convinced about assigning weights in goal programming. Here is an example formulation problem. What I tried: Let $x_j$ be the number of minutes for ad $j = R, T$ We want to ...
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$4$ or more type $2$ implies $3$ or less type $1$

I'm having difficulties with the logic with the last part of the reformulation part of the problem below. Let $x_i$ be the the number of ships of type $i$ to purchase. For $4a:$ (the ...
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1answer
21 views

Minimization problem with infinite variables and linear constraints

How can this minimization problem be solved? $$ \left\{\begin{matrix} \begin {aligned} &\sum_{i=1}^{\infty}P_i^3 \rightarrow min \\&\sum_{i=1}^{\infty}P_i=1 \\ &P_i\geqslant 0 \:for\: ...
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Time independent vs. time dependent lagrange multiplier

What are the differences between these two in applications? For example: $$max\sum_{t=0}^{\infty} \beta^t u(c_t)$$$$s.t.f(c_t,c_{t+1},x_t,x_{t+1})=0$$ What are the differences between: ...
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Where does the duality comes from in linear programing and can we get the optimal basis from it?

$$\begin{cases} \max & c^Tx\\ & Ax\le b\\ & x\ge 0 \end{cases}\Leftrightarrow \begin{cases} \min & y^Tb\\ & y^TA\ge c^T\\ & y^T\ge 0 \end{cases}$$ Then we come to the ...
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Nonlinear constraints replaced by parameters and estimated iteratively

I have an optimization problem with nonlinear constraints in the following form: $x + y + 0.5(x+y)^2-z = 0$ $s+(x+y)*t\ge M$ I linearize these constraint by replacing the nonlinear terms by ...
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4 views

Regularization as an alternating objective or combined objective

I have a "primary" task loss function $L=L1$ which I want to minimize. Adding a regularization term via $L=L1+\lambda L2$ can be thought of as "forcing" the optimal solution to be meaningful for a ...
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32 views

Linearize non-linear constraint [closed]

I have a problem which may be defined as: $$\max 5 x_{11} + 6 x_{12} + 2 x_{21} + 3 x_{22} \\ x_{ij}\in \{0,1\} \\ x_{11} + x_{12} = 1 \\ x_{21} + x_{22} = 1 \\ t_1,t_2 \text { integer} \\ (t_1 - ...
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8 views

Finding the lower bound of a linear program with the duality method

The issue I have some difficulties understanding the lower bound of a program when applying the duality method. It seems that it comes from $$c^T\underbrace{\le}_{x\ge 0\\y^TA\ge c^T} ...