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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Twilight Zelda Guardian Puzzle : Shortest Path Proof

I'm playing a video game right now and in it is a puzzle (see here). There are solutions to solving it (see here) on the Internet, but I'd like to know if this path is the shortest path (least amount ...
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2answers
22 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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What is an inner-outer iteration?

Inner-outer iterations are used in papers, for finding a stationary point of a system or in optimization. It is not clear, what is called an inner-outer loop though? Is it a nested loop where the ...
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10 views

Maximin optimization problem

I need to solve the following problem : Max.[ Min F(x,y ) ] where maximization is with respect to linear x , and minimization is with respect to non-linear y . The original problem had 6 ...
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9 views

Hierarchical Linear Programming

I am stuck with the following problem from research. For each time, $t$, I get a new data point $x_t$ and the current optimum value is a function of $\{x_t:t=1,2,\dots,T\}$ obtained by solving a LP. ...
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Minimising the surface area of a rectangular prism [Solution Verification]

A packaging company is going to make open topped boxes, with square bases that hold $100$ centimetres$^3$. What are the dimensions of the box that can be built with the least material?
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15 views

Find out the optimization type

I am formulating a problem and intend to solve it by optimization. Here is the current result: *Objective:*$\quad\min\quad c + f_1(x)x_1 + f_2(x)x_2$ Constraint: $\quad ax_1 + bx_2 <= d$ where ...
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26 views

Find the maximum value of $x^{\alpha}y^{\beta}$ subject to the constraints $x+2y \le 2$ and $x > 0$ and $y > 0$.

The Statement of the Problem: Given real numbers $\alpha > 0$, $\beta > 0$, $\alpha + \beta \le 1$, find the maximum value of $x^{\alpha}y^{\beta}$ subject to the constraints $x+2y \le 2$ and ...
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0answers
13 views

Heuristic Optimization

I am working on a project with the NMF and additional cost terms. Therefor I am looking for an optimal weight factor for the cost terms to maximize the result. Because it is NP-hard and needs some ...
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18 views

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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2answers
29 views

To show $f(x)$ has ONLY one Max in $x\in[0,1]$

I have function $$f(x)=\left(\frac{1-x}{2-x}\right)x^{p-1}~\text{where}~p>1;~x\in[0,1]$$ I want to show that $f(x)$ has ONLY one maximum in $x\in[0,1]$. I get the second derivative as ...
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35 views

How find this minimum

Help me! Let $x,y,z\ge0$ such that: $xy+yz+zx=1$. Find the minimum value of: $A=\dfrac{1}{x^2+y^2}+\dfrac{1}{y^2+z^2}+\dfrac{1}{z^2+x^2}+\dfrac{5}{2}(x+1)(y+1)(z+1)$ I found minimum value of $A$ ...
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0answers
15 views

Max-Min optimization problem with $N$ entries

I want to find the optimal $x$, say $x^*$, which maximizes the minimum of $N$ entries as given below: \begin{equation} \begin{split} &\max_{x}~\min ...
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1answer
19 views

Augmented Lagrangian Method for Inequality Constraints

Augmented Lagrangian Method can be used with inequality constraints. The question is how. One approach (according to Numerical Optimization Book by Nocedal and Wright; page 522), is linearly ...
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1answer
31 views

Minimum Volume of a circular, right cone, with a sphere inscribed in it.

Question: A sphere of radius $r$ is inscribed in a circular, right cone. What is the minimum radius and height of the circular cone? (Thus, volume) Because the answer would specifically ...
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1answer
74 views

No clear analytic method to prove unique maximum? ($2^{-x}+2^{-1/x}$)

Prove that $f(x) = 2^{-x}+2^{-1/x}$ has the unique local maximum $(1,1)$ for $x>0$. Do not use computer software. Proving that $(1,1)$ is a maximum is easy, but I'm having trouble with the ...
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22 views

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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0answers
14 views

Optimal Matching Distance

I'm stuck on problem II.5.9 from Bhatia's Matrix Analysis. The problem is as follows: Let $\{\lambda_1,\dots,\lambda_n\},\{\mu_1,\dots,\mu_n\}$ by two $n$-tuples of complex numbers. Let $$ ...
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0answers
12 views

Assignment problem, minization of the Standard Deviation

I have an assignment problem. So typically I need to find the optimal combination between two sets of parameters P, M. I know that the Hungarian Algorithm is often privileged for this kind of problem ...
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27 views

Formulate an optmization problem as a convex optmization problem

Let $P$ be a polyhedron, i.e. $P = \{ x \in \mathbb{R}^{n}\, |\,\, a_{i}^{T}x \leq b_{i} \}$. Define $R$ as the rectangle given by $\{ x \in \mathbb{R}^{n}\, \mid\, \, l \preceq x \preceq u \}$. Find ...
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1answer
24 views

Dimensions of a paddock (3 sides of a rectangle) to enclose maximum possible area

I need help with Qs 4, 5 and 6!! Three sides of a rectangular paddock are to be fenced, the fourth side being an existing straight water drain. If 1000m of fencing is available, what dimensions ...
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6 views

Spin-off of Scheduling Weighted Interval Problem

I'm trying to solve a problem in which, given a + sign shaped area of land (with no width) and a list of contiguous sections of the land (segments, T-shapes, smaller + shapes, etc), each with an ...
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2answers
30 views

Find Algorithm, given a list of arcs, that maximizes number that fit on a circle

I'm trying to find an optimal algorithm that, given a list of arcs $(x_i, y_i)$, where $x_i$ and $y_i$ are the starting and ending angle measurements of the arc in radians, maximizes the number of ...
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17 views

Algorithm to find value where complex numbers meet on the unit circle. [on hold]

I'm trying to find the value at which 4 points are meeting on the unit circle. These points are eigenvalues of the translation operator $T$. By varying $\lambda$ the eigenvalues change. Background: ...
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12 views

Linear programming (or possibly nonlinear) formulation

The problem is like this; The construction company is considering erecting three office buildings. The time required to complete each of them and the number of workers required required to be on the ...
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0answers
18 views

bound on Lagrange multipliers

Under what conditions is it possible to bound the Lagrange multipliers in a given optimiztion with constrains problem?
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3answers
64 views

Question about maximizers and trig

Hi there I have a quick question about the following Consider the simple maximization problem of $$f(x,y)= \frac{x}{1+x^2+y^2}$$ It can be easily seen from analysis of critical points obtained from ...
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2answers
30 views

compute the smallest affine subspace containing $S$, where $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ is a set of vectors in $\mathbb R^3$

I've started to study convexity to enchance my optimization skills. Given a set $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ of vectors in $\mathbb R^3$ an exercise asks to compute the smallest affine ...
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converting a equation to convex form which can be given to cvx solver to solve it. [on hold]

Can anybody tell me how to convert this to quadratic programming format so that CVX could solve it...?? I am not asking the whole solution but need only conversion. objective is:- minimize {sum ( ...
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1answer
29 views

A problem on finding the nearest points to the origin on the intersection of two surfaces

Suppose we are to find the points nearest to the origin on the curve of intersection of the two surfaces $g^{-1}_{1}\{ 0 \}$ and $g_{2}^{-1}\{ 0 \}$, where $g_{1}: (x, y, z) \mapsto x^{2} - xy + y^{2} ...
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1answer
31 views

Envelope Theorem and Static Optimization

The Statement of the Problem: For fixed $r \gt 0$ and $m$, find the maximum value of $1-rx^2-y^2$ on the constraint set $x+y=m$. Find the value function $f^*(r,m)$ and compute $\frac{\partial ...
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1answer
70 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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17 views

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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2answers
20 views

Why is the gradient of the objective function in the Lagrange multiplier theorem not $= 0$?

A special case of the Lagrange multiplier theorem may be stated as: Let $S, T \subset \mathbb{R}^{n}$ be open. Let $f: S \to \mathbb{R}$ be differentiable on $S$ and $g: T \to \mathbb{R}$ ...
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1answer
27 views

Non-uniqueness of worst-case (max-min or min-max) optimization

I have a worst-case optimization problem, where i want to maximize the minimum from the uncertainty set (uncertainty is given as an ensemble of 100 realizations, so an ensemble based approach). It is ...
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14 views

Mixture of Maximum Entropy and Minimum Cross Entropy?

Assume you have a discrete prior distribution on a set of points $ P(X=\{0,3,5,6\}) = (.40,.30,.20,.10)$ $E[X]=5/2$ And you want to create a new distribution, $Y$, on $\{0,1,2,3,4,...\}$ using the ...
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Positive solutions to $A^T A x \geq 0$ [on hold]

Find a positive solution $x$ to the linear inequality $A^T A x \geq 0$. $A$ is an arbitrary matrix. I was wondering if there is a general solution. EDIT: One special solution is when $A^TA$ is row ...
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1answer
26 views

What would be the basic solution of this maximization problem? [on hold]

Maximize $P=40x_1+50x_2$ Subject to $x_1+6x_2 \leq 72$ $x_1+3x_2 \leq45$ $x_1, x_2 \geq0$
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minimising multivariate quadratic function over integer variables

I have a quadratic function $x_1^2+x_2^2-(u_1x_1+u_2x_2)^2$ which I need to minimise over integer $x_1$ and $x_2$; also, the coefficients $u_1,u_2<1$. In other word, assuming coefficients ...
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1answer
23 views

An LP problem from David G. Luenberger's Linear and Nonlinear Programming book

Could someone help me to solve the following problem? A class of piecewise linear functions can be represented as $f(x) = Maximum (c_{1}^Tx+ d_{1}, c_{2}^Tx, \cdots, c_{p}^Tx + d_{p})$. For such a ...
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2answers
41 views

How to minimize $w^{T}Aw$?

$A$ is $n\times n$ matrix. Find a $w$ ( $n$-dimensional unit vector) which minimizes this function. By $w^{T}$, I mean $w$-transpose. I understand there would be non-linear optimization techniques ...
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22 views

maxima minimum problem 7 [closed]

Find the maximim and minimum value of the function $f(x,y)=(x+1)^2+y^2$ on the part of the graph of $y^2-x^3=0$ from $(1,-1)$ to $(1,1)$ Can someone help?
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1answer
27 views

maxima minima optimization problem

My problem is Find the greatest and least distance of the surface $6x^2+4xy+3y^2+14z^2=14$ from the origin. I know that mathematical model of problem is $f(x,y,z)=x^2+y^2+z^2$ subject to ...
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28 views

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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1answer
45 views

A maximization problem parametrized by a function

Let $f$ be a smooth positive monotonically increasing real function which is defined and finite in $[0,1]$, and define the following two quantities (see the figure below): $F=\int_{x=0}^1{f(x)dx}$ = ...
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1answer
30 views

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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2answers
48 views

Shortest line segment that is cut off by the first quadrant and passes through a given point

Let $a$ and $b$ be two positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point $(a,b)$. I attempted to let the ...
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0answers
17 views

Minimizing l2-norm of convolution (Perron-Frobenius theorem)

I need to minimize the $||\mathbf{h}*\mathbf{x}||_2$, where $\mathbf{h}$ is a given non-negative vector, and $\mathbf{x}$ should be a compactly supported non-negative vector. In the matrix form, this ...
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16 views

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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2answers
21 views

The optimization problem with max [closed]

Given $(m\times n)$-matrices $A=(a_{ij})$ and $B=(b_{ij})$, and a vector $c=(c_1, c_2, \ldots, c_m)$; and $\underline{x},\overline{x},\underline{y},\overline{y}$ are real numbers such that ...