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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To show that $f(y)$ has only one maximum in $y\in[0,1]$

I have function $$f(y)=\frac{1}{2} y \log \left(\frac{a^2 b \left(\frac{2}{y}-2\right)}{a b \left(\frac{2}{y}-2\right)+a+1}+1\right)$$ where $a,b>0$ and $y\in[0,1]$. I want to show that $f(y)$ ...
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Optimization involving convex-concave function

Let $f(x,y)$ be a function defined on $[0,1]^2$ and define \begin{align} g(a,b) = f\left(\frac{a+b}{2}, \frac{a-b}{2}\right) \end{align} where $a$ and $b$ are such that $\frac{a+b}{2} ...
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Can I clamp singular values of $3\times3$ matrix without effectively computing SVD?

I have a $3\times3$ matrix $A$, and compute its SVD $U \Sigma V^\star = A$. I clamp the singular values in $\Sigma$ to some small range (e.g. $[0.5, 1.5]$ ) and reconstruct matrix $\widetilde{A}=U ...
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38 views

Non linear programming problem using Kuhn-Tucker method

Solve using Kuhn-Tucker method $ z=x_1^2+x_2^2 $subject to i) $ x_1+x_2\le 4$ ii)$ 2x_1+x_2\le 5$ where $ x_1\ge 0 ,x_2\ge 0$
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Has extremum or not?

I'm learning calculus and I have to do with functions $x^2\sin(\frac{1}{x})$ where x!=0 and 0 when x=0 and $x^3\sin(\frac{1}{x})$ where x!=0 and 0 when x=0 If I computed this well, both of them have ...
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16 views

Determine min and max of the following expressions

Let $a_1,\; ...\;, a_n$ and $b_1,\; ...\;, b_n\in \mathbb R$ be positive real numbers. Find $$ max \;(a_1x_1 + a_2x_2 + ... + a_nx_n) $$ and $$ min \;(a_1x_1 + a_2x_2 + ... + a_nx_n) $$ over $x_1, ...
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29 views

Multiple optimal solutions / LP

In the optimal primal simplex tableau, if we have a non-basic variable with a reduced cost of zero, can we say for sure the primal has multiple optimal solutions? Or can the same thing also happen ...
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33 views

tangent cone to the set

I'm supposed to solve this problem: Let us consider the set $M=\{(x, \sin{x}):x\in\mathbb{R}\}\cup\{\big(\cos(x)-1,x\big):x\in\mathbb{R}\}$ The question is to find the tangent cone to the set $M$ in ...
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30 views

Derivative vector and Hessian for maximization

I'm having some troubles regarding maximization of approximated utility. I want to use the Newton method, but in order to do so I need the derivative vector and the Hessian matrix (I will be ...
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17 views

Saturation Curve

I have an expression which is $\displaystyle\frac x{x+40}$. I'm trying to find a point indicated in the graph. As you can see I drew 2 lines, one tangent to the region which it saturates, the other ...
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54 views

Must Number of equality constraints and decision variables be equal?

Must Number of equality constraints and decision variables in an optimization problem be equal ? If not, how can I solve the equality constraint equations with a solver e.g. ...
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37 views

Optimization of sum of logs

I have an optimization problem of the form $$\operatorname*{argmax}_{\mathbf{w}} \sum_i \log(1 + \mathbf{w} \cdot \mathbf{k_i})$$ given some set of vectors, $\mathbf{ \{k_i\} }$. I have tried both ...
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45 views

proving that the shortest line conntecting a point and a line will be perpendicular to that line

So I have a problem for my final math project that I've been fiddling with for hours without success. I have to use calculus to prove that the shortest line connecting a point to a line will always be ...
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How to solve this kind of Lagrangian function?

Suppose $\mathbf{a} = (a_{0}, \dots, a_{N-1})$ and $\mathbf{b} = (b_{0}, \dots, b_{N-1})$ with $a_{i}\geq0$, $b_{i}\geq 0$. I would like to minimize $$-\sum_{i=0}^{N-1}a_{i}b_{i}$$ subject to ...
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36 views

Finding global maximum

I have a program which must quickly find $x$ and $y$ where $x,y\in\mathbb{N_0}$ which correspond with maximum value of a function: $$f(x,y)=\frac{\sum_{i=0}^{|b|-1}{|b_i ...
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2answers
47 views

Simple Lagrange Multiplyers Problem

Can anyone please help me with the following: Find the stationary values of $u=x^2+y^2$ subject to the constraint $t(x,y) = 4x^2 + 5xy + 3y^2 = 9$. The answer is given as $u = 9$ and $x = \pm ...
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Homework help on eigenvalues function minimization

so I actually have two separate questions which are homework bonuses for my numerical methods course. Unfortunately, because of the time of the semester, our TAs are not available so I don't have many ...
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28 views

How can I solve the following exercise

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
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Binary depending on the sign of another variable

I'm writing a mixed integer linear problem, where I have an indicator function in the objective function counting the instances of negative values of a decision variable. I thought of defining a ...
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Quadratic programming with constrained number of free variables

I started with a (positive-definite) quadratic programming problem subject only to a single equality constraint. i.e. $$ f(x)=x^{T}Qx+c^{T}x $$ $$ s.t. x_1+x_2+x_3+...+x_n=y $$ I now have to find ...
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Say optimal solution to the primal is degenerate. Does it hold that optimal solution to dual not unique?

I think it's supposed to be that existence of a degenerate and unique solution of the primal implies multiple solutions to the dual, according to this book (pages 141-145, proof of Theorem 4.5). In ...
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Definition functions, integrals on $\mathbb R^{|N|}, \mathbb R^{\mathbb R}$

Is there a standard/reasonable way of defining functions on the sets $\mathbb R^{|\mathbb N|}, \mathbb R^{\mathbb R} $. How about defining integrals over these sets? I guess a function on $\mathbb ...
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41 views

Shortest smooth paper Möbius Strip

I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What ...
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Lipschitz Number in Gradient Descent

During gradient descent, if an objective function's value is greater than the previous iteration, would use of an orthogonal vector to the update vector be advantageous? Regarding trust regions, the ...
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32 views

Maximum of norm

Given a matrix $A$ with $N$ rows and $d$ columns, I would like to prove (or disprove) the following. Let $q(f)=\|(\begin{pmatrix} ...
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40 views

Gluing two strongly convex function

Definition: We call $f:\mathbb{R}^n\rightarrow \mathbb{R}$ a $\lambda$-strongly convex function iff for every $x,y\in \mathbb{R}^n$ and $t\in[0,1]$ it follows $$f(tx+(1−t)y)\leq ...
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23 views

How to model a multiobjective problem with a large dataset?

I have a large dataset of businesses (around 5k venues with distance from a predefined point, average price and service quality rating) and I need to create the objective functions to minimize the ...
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33 views

Do quasi-Newton methods check the second-order optimality condition?

I have a practical question about quasi-Newton methods. In quasi-Newton methods, Hessian matrix is approximated. It seems to be impossible for them to check the second optimality condition. In ...
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Grouping constrained optimization

I am looking for an efficient solution to solve the following problem. Can anybody help? S is a finite set of elements $k_i$ V is a subset of S, e.g. $v_4$={$k_1$,$k_3$} E is a finite ensemble of ...
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Why Was Backprop Invented?

I'm currently researching artificial neural networks and I keep wondering why do we use "backpropagation" to train a neural network. An ANN is basically just a very large and complex function ...
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Lagrange Multiplier for clustering with size constrains

I'm trying to solve a clustering problem with size constrains. Minimize $J=\sum_{i=1}^c\sum_{j=1}^n {{u_i}_j}^2{d_i}_j$ Subject to $\forall 1\le j\le n : \sum_{i=1}^c {{u_i}_j}=1$ and ...
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Optimal strategy in the following game:

In this game, 12 hidden D6s are rolled and summed. The player is given the total of the rolled dice. The player will then guess a number from 1 to 6. If there is a unrevealed dice with that number, ...
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Does a larger change in $R^2$ indicate a given parameter is “more important” in model fitting?

I have a set of experimental data, and a given model which is supposed to fit that data. Let's say this model has $n$ parameters. I need to write a parameter estimation algorithm to determine the ...
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59 views

Finding $ \max_{x \in [2,4]} \left| 2 x \cos(2 x) - (x - 2)^{2} \right| $.

This is a problem taken from Burden’s and Faires’ Numerical Analysis. Define $ f: \Bbb{R} \to \Bbb{R} $ by $$ \forall x \in \Bbb{R}: \quad f(x) \stackrel{\text{df}}{=} 2 x \cos(2 x) - (x - 2)^{2}. $$ ...
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connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian

In the context of solving linear programs, the big-M method refers to adding additional variables to the problem such that there is, as far as I understand it, a trivial basic feasible solution. In ...
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60 views

Implementation of Lagrange Multiplier to solve constrained optimization problem.

I'm trying to solve an optimization problem. I have a list of around 4000 geo coordinates data, and I want to cluster them into 30 groups based on the distance, so that the closer properties belongs ...
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3answers
50 views

Finding the absolute maximum of the following 3d function

$ f(x,y) = \frac{(\lambda_1x+\lambda_2y+\lambda_3)^2}{x^2+y^2+1} $ I know that the function looks like some deformed dorito chip depending on the lambda values. That is about as far as I've gotten. ...
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Feasible solution with positive $m+1$ components

Can anyone give me a suggestion? Let \begin{equation} \min \hspace{0.3cm} \{c^Tx: \text{ s.t. } Ax = b, x \geq 0 \} \end{equation} Suppose that $x$ is a feasible solution to the previous LP, with ...
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How to prove a solution is indeed a constrained minimum?

I'm reading the following example on Heath's Scientific Computing (page 266, second edition if anyone has it). "Minimize $f(x_1,x_2)=2\pi x_1(x_1+x_2)$ subject to $g(x_1,x_2)=\pi x^2_1x_2-V$" ...
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89 views

Dimension of garden to minimize cost

Math question: A homeowner wants to build, along her driveway, a garden surrounded by a fence. If the garden is to be $5000$ square ft, and the fence along the driveway cost $6$ dollars per foot while ...
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Epsilon constraint method - Pareto optimal solution representation

There's a course that I do remotely and I have a homework question which I have no idea how to answer. I did look up a lot in google and did not find any good examples - only loads of information and ...
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Minimizing Sum of Least Squares in Matlab

I am working on this minimization problem for image warping that I want to solve in Matlab: Each feature $p$ can be presented by a 2D bilinear interpolation of the four vertices $V_p = [v_p^1, ...
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If I want to learn mathematical optimization where should I start?

I'm working in the area of transportation engineering, to be specific, mostly involved in management. I read some papers in Transportation research part x, and something like that, and noticed that I ...
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How to find optimal subset $I$ such that $(\sum_{i \in I} a_i)^x / \sum_{i \in I} b_i$ is maximized?

Suppose we are given pairs $(a_i,b_i)$ of positive numbers and $x \geq 1$. The goal is find the optimal subset of indices $I$ that maximizes: $$\frac{(\sum_{i \in I} a_i)^x}{\sum_{i \in I} b_i}$$ ...
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what is the optimal matrix solution for this problem?

i want to maximize this objective function: $$j(W)=\frac{\mathrm{tr}(WAW^T)}{\mathrm{tr}(WBW^T)}$$ Where $W$ is a $f\times m$ matrix and the matrices $A, B$ have size $m\times m$ and $\mathrm{tr}$ is ...
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Cross-entropy minimization - equivalent unconstrained optimization problem

I'm looking at this paper "An Alternative Method for Estimating and Simulating Maximum Entropy Densities" ...
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Maximizing the frobenius norm subject to constraints $\underset{\mathbf{S}}{\text{maximize }} \|\mathbf{S}\|_F^2$

IF $\mathbf{X=AS}$ where $\mathbf{X} \in R_+^{n \times m}$, $\mathbf{A} \in R_+^{n \times r}$ are known variable and $\mathbf{S} \in R_+^{r \times m}$ is unknown variable, How to solve the below ...
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Semidefiniteness of the Hessian and optimization

This question is for sure a duplicate, but different users seem to give different answers. The question is: suppose you find that the Hessian matrix for a function $f(\textbf{x})$ is semidefinite ...
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28 views

Finding the critical points of $f(x,y) = x y^2 - x^2 y + x y$

Trying to find the critical points of $f(x,y) = y^2x - yx^2 + xy$. I took partial derivative with respect to x, so $F_x = y^2 - 2xy + y$ $F_x = y(y - 2x + 1)$ Then with respect to y, $F_y = 2xy - ...
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Recovering the optimal primal solution from dual solution

I'm having trouble finding the optimal primal solution of a particular problem from its dual solution. Primal: $\texttt{Maximize} \ \ 10 x_1 + 24 x_2 + 20 x_3 + 20 x_4 + 25 x_5$ Subject to $x_1 + ...