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 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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Augmented Lagrangian Method for Inequality Constraints

Augmented Lagrangian Method can be used with inequality constraints. One approach (according to Nocedal book page 522), is linearly constrained Lagrangian. Description is shown in the attached image. ...
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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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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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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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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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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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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
12 views

Dimensions of a paddock (3 sided 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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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
28 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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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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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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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
62 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
27 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
25 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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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
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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
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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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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
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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
22 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
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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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maxima minimum problem 7 [on hold]

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
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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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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
43 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
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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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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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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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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
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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 ...
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1answer
25 views

Maximization of a function in an interval

I am writing a computer program where I have $x$ real positive varying in the domain $[\sqrt{U}, U]$. I want the value of $x$ which maximizes: $$ (1+ \sqrt{U}) - \frac{\sqrt{U}-1}{U-\sqrt{U}} x - ...
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34 views

Optimization on fixed sum

Consider this following scenario. Suppose I have $N$ cents, and I want to dispatch these money to $n$ people, each got $x_i$ cents. In order to simplify this problem, we assume the cents are ...
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2answers
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A differential maximization problem

OK, I know how to solve maximization problems on numbers, and I know how to solve differential equations which are equations on functions, but how do I solve a 'maximization problem' over functions? ...
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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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Maximizing profit (dynamic programming)

I'm looking at a dynamic programming question and can't figure out how to solve it. The question is listed at the following website (question number 19, towards the bottom). ...
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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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Can $\sin (x)$ be represented as difference of two convex functions?

In my optimization homework, I am supposed to prove that every differentiable function with Lipschitz continuous gradients can be represented as difference of two convex functions. I think I have come ...
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Critical point outside of domain when finding the intervals on which a function is increasing and decreasing.

I have this function: f(x)=x^(1÷3) × (x+8) I'm trying to find the intervals on which the function is increasing and decreasing. Then, I am to find the local extrema. I've done this: f'(x) = ...
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Maximizing revenue [closed]

A coffee wholesaler sells two types of beans. Arabica beans that sell for $ \$8 $ a pound and Selecto beans that sell for $ \$24 $ a pound. The Arabica beans cost $\$1$ per pound to store and the ...
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train cost optimization problem

Fuel cost for operating a train is proportional to the square of the speed, and is Rs.50 per hr when the speed is 20 mph. Other charges, such as labor, for example, put together is Rs.200 per hr. The ...
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5answers
123 views

Maximum value of $ x^2 + y^2 $ given $4 x^4 + 9 y^4 = 64$ [closed]

It is given that $4 x^4 + 9 y^4 = 64$. Then what will be the maximum value of $x^2 + y^2$? I have done it using the sides of a right-angled triangle be $2x , 3y $ and hypotenuse as 8 .
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Dual norm of quasi norms

The dual norm $\Omega^*$ of the norm $\Omega$ is defined for any vector $\mathbf{z} \in \mathrm{R}^N$ by \begin{equation} \Omega^*:= \underset{\mathbf{x} \in \mathrm{R}^N}{max } \quad \mathbf{z}^{T} ...