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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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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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
20 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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0answers
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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
24 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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0answers
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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
68 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
19 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
14 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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11 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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39 views

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
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
39 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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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
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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0answers
24 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
42 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
45 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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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
21 views

The optimization problem with max [on hold]

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
24 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
25 views

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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1answer
35 views

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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2answers
52 views

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 [on hold]

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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1answer
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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
119 views

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

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} ...
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1answer
38 views

Given $R \in \mathbb R$, choose $a,b,c$ from discrete set so that $a^{-1} + b^{-1} + c^{-1} \approx R^{-1}$

I am working with the following equation (parallel resistors): $\frac{1}{R_g} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$ The values of $R_1, R_2$ and $ R_3$ are discrete - lets say 256 steps ...
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Optimal $A\in \Sigma$ that maximizes an objective

Let $([0,1],\Sigma, \lambda)$ be a probability space. For any given $B\in \Sigma$, $K\in [0,1]$ and $f\in L^2(\lambda)$ with $f(x)\in[0,1]$ for all $x $, $$\max_{A\in \Sigma}\int_A f(x) d\lambda(x)- ...
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Matrix norm optimization problem : $\min_{\textit{ }x} \| A x B \|_4$, $x$ in the “unit” circle

Bonjour, Let $A$, $B$, $C$ and $D$ complex matrices. Is there a way to find a matrix $x$ (edit: non trivial) as: $\min_{\textit{ }x} \| A x B \|_4^4$ Or, more complicated, $x$ as $\min_{\textit{ ...
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calculus optimization help

Minimum Cost A storage box with a square base must have a volume of 80 cubic centimeters. The top and bottom cost 0.20cents per square centimeter and the sides cost 0.10cents per square centimeter. ...
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1answer
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How to optimize these parameter

How to optimize the following respect to lambda1 and lambda2: $\sum_{i} f(i) * log(\lambda_1 g(i) + \lambda_2 h(i))$ f(i), g(i), h(i) are known funtions Find lambda1 and lambda2 that satisfy ...
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1answer
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$\arg\max$ of an increasing function grows as the region grows.

$f_1,\dots,f_N:\mathbb{R}^+\rightarrow\mathbb{R}^+$ are strictly increasing, bounded functions whose derivatives monotonically decrease to $0$ as their argument increases. (Picture the shape of the ...
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1answer
36 views

Given the set of all polygons with m sides and perimeter 1, why is there an element with maximal area?

The set of all polygons with $m$ sides and perimeter $1$ has an element with maximal area. I read this fact in a book, and the reference was in German. Does anyone here know? I know how to ...
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1answer
31 views

How can I project a matrix on the set of symmetric positive definite matrices with trace 1?

Given a square matrix $A \in \mathbb{R}^{n \times n}$, I need to compute $$ \min_{X \in \Omega} \lVert A - X\rVert^2$$ where $\Omega = \{X \in \mathbb{R}^{n \times n} |\, tr(X) = 1, X \text{ is ...
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Multivariate Optimization [on hold]

I'm currently in a multivariate optimization course and literally have no idea what I'm doing. If someone could help me with this problem or at least tell me where to start I'd be eternally grateful. ...
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2answers
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Is there a name for this modified newton's method?

I know that Newton's method has the following formula: $$f_{t+1} (x)= f_{t}(x)-f'(x)/f''(x)$$ The source code at the end of the post seems to use the following construction instead: $$f_{t+1} (x)= ...
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How to convex problem with multiple constraints? $\underset{\mathbf{A}}{\text{minimize }} \|\mathbf{A}\|_F^2$

IF $\mathbf{X=AS}$ where $\mathbf{X} \in R_+^{n \times m}$, $\mathbf{S} \in R_+^{r \times m}$ are known variable and $\mathbf{A} \in R_+^{n \times r}$ is unknown variable. here $R_+$ denotes the set ...
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Non-convex function with global minimum [duplicate]

I am working on a complicated objective function which I suppose is not convex. But when I use a global optimization tool that can find all its local minimums, it will always converge to the same ...
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1answer
26 views

Convex optimization problem: linear equality and inequality constraints

When linear equality constraints can be converted in an inequality constraints for a strongly convex optimization problem? I mean, I got the same solution for both the following problem: 1) $\min_x ...