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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Linear programming problem neither max nor min

Heres the actual question: television provider broadcasts two movie channels, A and B. Channel A broadcasts 1 romantic movie, 3 action movies and 3 comedies per month at a cost of 50 Euro. ...
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Determine the points where $f$ is has a local minimum/maximum. Multivariable calculus question.

This is not homework, but it is in my book and I find it hard to solve: Determine the points where $f$ is has a local minimum/maximum. Determine if it strong/weak and absolute/relative and ...
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509 views

Cost minimization problem

The problem is as follows: A firm uses $k$ units of capital and $l$ units of labor to produce $(k^{\alpha}l^{1-\alpha})^{1/\beta}$ units of output, where $\alpha$ and $\beta$ satisfying $0 < ...
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Optimal Basic Feasible Solutions

In linear programming, is it true that you can only have at most 2 optimal basic feasible solutions? If so, why?
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70 views

Is there an explicit solution to: $\arg \min mn : mn \geq k, l_0 \leq n \leq l_1$?

Is there an explicit solution or a fast algorithm to compute: $$\underset{(m, \ n) \in \mathbb{N}_{+}^2}{\arg \min} \ mn \ : \ mn \geq k,\ l_0 \leq n \leq l_1$$ for given constants $k, l_0, l_1 \in ...
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Optimization: Minimize cost of pipeline

A small resort is situated on an island off a part of the coast of Mexico that has a perfectly straight north-south shoreline. The point P on the shoreline that is closest to the island is exactly 6 ...
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867 views

Maximize $f(x) = x^3-3x$ subject to constraints

I would like to understand more about how to maximise functions of one variable subject to constraints. How can you find the maximum value of $f(x) = x^3 - 3x$ subject to $x^4+36 \leq 13x^2$? The ...
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In this situation, how to decrease the payout amount based on the “weighed” constituent parts?

Couldn't think of a better way to explain it in the title, my bad. This is a problem I'm running into in a programming project. Here's the situation: Say a partner is due to be paid $1000 for ...
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1answer
117 views

Maximizing fire-breathing power of multi-headed dragons

Dragons are gathered up on a battlefield. Certain dragons are chosen in order to provide maximum fire breathing power. A dragon can have any number of heads. The only rule is that no more than $1000$ ...
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137 views

Finding absolute max and min values of function

Function given as $f(x,y) = 3x^2 + 2xy^2$. If $(x,y)$ lies in the region inside including edges of the triangle in the first quadrant given by $x\ge0, y\ge0, y\le2-x$. Reduce $f$ to a single variable ...
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160 views

HELP please with Optimization with constrain using lagrangian

I am reading this book on optimization and they present the following problem: Lisa wants to maximize her utility U(q1,q2) subject to a budget constrain, budget constrain is $p1*q1+p2*q2=I$. Ok , I ...
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508 views

Finding the length and width of a house that maximize its area

A house is built in the shape of a rectangle, with $3$ rectangular interior sections separated by parallel walls, using fencing. The owner has $900$ feet of fencing, and he wants to enclose the ...
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116 views

Simplex method: Utter, extreme confusion

We want to maximize $ z = 30x_1 + 20x_2$ with $$2x_1 + x_2 \leq 140$$ $$x_1 + 2x_2 \leq 160$$ $$x_1 + x_2 \leq 90$$ $$ x_1, x_2 \geq 0$$ So my book says the first step is writing these to ...
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1answer
39 views

Existence of Minimum Value

Assume $x\ge0$, show that the function $f(x,y)=(2xy+y^2)e^{-x}$ has a minimum value. Note that actual minimum value is $-4e^{-2}$.
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76 views

Tricky algebra for minimization

Find the local minimum for $f(x, y) = 2x^4 + y^2 - 4xy + 5y,\:x,y \in \mathbb{R}$ find the local minimum. Okay this seems easy enough, the necessary condition dictates that candidates are of the form ...
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Max and min value of $7x+8y$ in a given half-plane limited by straight lines?

So, there are four inequalities: $$\begin{eqnarray*} y &\geq &-3x+15; \\ y &\leq &-11/3x+56/3; \\ x &\geq &0; \\ y &\geq &0. \end{eqnarray*}$$ If we draw all those ...
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479 views

Maximize the product of linear functions

Suppose $f(x,y) = \prod_{i=1}^n (a_ix+b_iy)$ where $n$ is a constant larger than 500, and $a_i>0$, $b_i>0$ are known coefficient. There is only one global maximum. What's the most efficient ...
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403 views

Derivative inside an integral

Assume that I have an integral $$ I=\int_\Omega f(\omega)g(\omega)d\omega, $$ where $\Omega$ is a measure space and $\omega\in \Omega$. What is $$ \frac{\partial I}{\partial f(\omega)}? $$ i.e. I ...
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81 views

reformulating $\arg\min |x|^{1.5}$

I've asked this question. Essentially I have the same question for problem $\arg\min |x|^{1.5}$. We can consider next variant $\arg\min t^{1.5}$ subject to $x\le t$ and $-x \le t$. But $t^{1.5}$ is ...
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43 views

optimization help please

I have a test in a couple of hours and i dont know how to do volume questions please help me out with thiss pleasee The base of a particular solid S is a circular disk ( a "filled in circle") with ...
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322 views

minimizing a norm and a linear function

Let $y,\lambda\in\mathbb{R}^n$. I want to minimize the following with respect to $y$. $$ f(y)=||y|| + \lambda^Ty $$ where $||y||$ is the Euclidean norm. I first take the derivative of the function and ...
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60 views

What mathematical operations are used to determine maximum resource utililzation?

I'm playing a game and would like to leverage mathematics to use optimal resource utilization. There are three troops I can build (ground, air, and horse). There are also three resources (wood, ...
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73 views

Reformulation of BQP to SDP

I run into the following reading some optimization papars: $$\min_x x^TAx $$ where $x\in\{-1,1\}^n$ and $A\in S_n$, Is equivalent to $$ \min <X,A>$$ s.t $diag(X) = (1,1,...,1)\;\; rank(X) = 1$. ...
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31 views

My numbers don't feel right on this maximization question.

If Jack is going to construct a rectangular dog pen and divide it into 3 equal subpens (with two fences inside parallel to one side). What is the maximum area of the overall pen if he only has 240 ...
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131 views

Is this use of the simplex method correct?

I am trying to implement a simplex algorithm for solving LP task. I will post the question and my solution as well - what I need to know is whether my solution is correct, thanks in advance! ...
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476 views

Maximizing the Determinant Function

Let $M_{n}$ denote the set of $n\times n$ real matrices. Let $c>0$ be a real number and denote by $X_1,X_2,...,X_n$ the lines of the matrix $X\in M_n$. Let $\|X_i\|$ denote the euclidian norm of ...
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480 views

Using Lagrange multipliers for restricted extrema

Consider the function $f(x,y) = x^2 + xy + y^2$ defined on the unit disc $D = \{(x,y) \mid x^2 + y^2 \leq 1\}$. I can not simplify the equations to the point where I find a constant for the lagrange ...
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40 views

Approximation in $L^2$

Let $G$ be a domain assumed smooth enough. I want to show that the mean value $m$ is minimizing $ m \rightarrow \| f-m\|_{ L^2(G)} $ for $ f \in L^2(G)$. Is it unique? Is it allowed to derive under ...
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856 views

Did I minimize the cost correctly?

I have a word problem that reads as so A farmer wants to fence a rectangular part of his land (30,000 square feet). The fenced area is to have one border shared with a neighbor which he wishes to be ...
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vector optimization

Trying to solve for (vertical) vector $β$ of length $n$, that maximizes scalar function $f(β)$ $$f(\beta) = \frac{\beta^T \mu}{\sqrt{\beta^T M \beta}}$$ where $μ$ is a (vertical) vector of length ...
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Finding local maxima, minima, and saddle points of $f(x,y)=xy + \frac1x +\frac1y$

Can anyone show me whether my answer below is correct and complete? Specifically, I am not sure whether or not I defined the extrema in explicit-enough terms. Also, the graph of the function using ...
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Linear least squares with non-negativity constraint

I am interested in the linear least squares problem: $$\min_x \|Ax-b\|^2$$ Without constraint, the problem can be directly solved. With an additional linear equality constraint, the problem can be ...
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A sufficient condition for a unique maximum of the product of two concave functions

Given two concave functions $f(x)$ and $g(x)$, what conditions in terms of these functions can ensure that $h(x)=f(x)g(x)$ have a unique maximizer on an interval $[a,b]$ for $a<b$?
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2nd Order Optimal Control Problem

I'm working on a homework problem in optimal controls and my plant model is described as: $$\ddot{x}(t) = u(t)$$ The performance index (cost function) is described by: $$J = 1/2\int_0^5u^{2}(t)dt\,$$ ...
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204 views

Economic Analysis / Minimization Problem

I am studying and going through some old exams for a microeconomic analysis class. I am just looking for some clarification regarding one of the answers given. The question is as follows Suppose ...
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796 views

Bell-shaped polynomial over a limited domain

The function $f(x) = e^{-x^2}$ has a bell-shaped peak at $x=0$ and then approaches an asymptote at $y=0$. I need to achieve a similar result, but with a polynomial function. I can use a series ...
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Why is ROC analysis not used in optimization problems?

In machine learning and applied fields of statistics, receiver operating characterization (ROC) analysis is commonly used to select optimal algorithms/models. However, at a lecture I once attended on ...
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Verifying an error estimate on a multidimensional function using its jacobian

This is taken from Nocedal & Wright's Numerical Optimization 2nd edition, pg 279 Theorem 11.3: Suppose $r:R^n\rightarrow R^n$ is continuously differentiable in a convex set $D\subset R^n$. Let ...
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599 views

Understanding how to state the Karush-Kuhn-Tucker Conditions for a given problem

I'm trying to understand an example given by Nocedal & Wright (1999), pg 329, Example 12.4. According to a definition given earlier in this book: At a feasible point x, the inequality ...
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638 views

Determine the nature of a critical point (Lagrange multipliers)

Let $F \colon \mathbb R^2 \to \mathbb R$ be the function $$ F(x,y):=xye^x + ye^y - e^x+1 $$ and denote with $C$ the set of zeroes of $F$, i.e. $C:=\{(x,y) \in \mathbb R^2 : F(x,y)=0\}$. Let ...
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145 views

Lower bound of $J=\frac{x^TAx}{x^TBx}$

Consider two symmetric positive semi-definite matrices $A, B \in \mathbb{R}^{n\times n}$. Suppose that $A$ and $B$ have the same null space $\mathcal{N}\subset \mathbb{R}^n$. Now consider the ...
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Finding figure of least perimeter

I was stuck (not exactly) at this problem for about a month and finally decided to ask my doubt here. This problem is: Given a family of parallelograms '$R$', all of which are on equal bases ...
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616 views

Bilinear Optimization Problem

How could I solve the following optimization problem using MATLAB or an other way? Given ${E}^{1}, {A}^{21}, {A}^{22}, {C}^{1}, {A}^{12}$ $ \underset{{C}^{2}, {E}^{2}}{min} {\left \| {C}^{2}{E}^{1} ...
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What is a good technique to decide step size in sub-gradient method for dual decomposition?

I am looking at the following paper to implement dual decomposition for my algorithm: http://www.csd.uoc.gr/~komod/publications/docs/DualDecomposition_PAMI.pdf On Pg.29 they suggest setting the step ...
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How to compute the min-cost joint assignment to a variable set when checking the cost of a single joint assignment is high?

I want to compute the min-cost joint assignment to a set of variables. I have 50 variables, and each can take on 5 different values. So, there are 550 (a huge number) possible joint assignments. ...
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Petri net analysis.

I have problems with this exercise. First: can the token in place $p_1$ to enable the transitions $t_2$ and $t_3$? The place $p_1$ has a single token, I think it fails to enable $t_2$ and $t_3$. Any ...
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Maximizing the time we reach to a threshold in a series of numbers

I have a problem and I really don't know what kind of mathematical method should I apply to solve or model my problem. I would be thankful If anyone can give me some answer or help. Suppose we have ...
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1answer
4k views

Find region of integration where triple integral has maximum value

I have to find the region $E$ where $$ \iiint_E (1-x^2 - 2y^2 - 3z^2) dV $$ has maximum value, but I'm not sure how to start. I was thinking of getting the derivative of the integral and then ...
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135 views

Number of Lagrange Multipliers

Suppose we are looking at the following: $$ \text{minimize} \ f(x) = x^2+y^2 \\ \text{subject to} \ \ x+y-2 \geq 0$$ Would there only be one Lagrange multiplier corresponding to the single ...
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Optimization problem defined by polynomials only always leads to algebraic solutions?

Let $\Omega$ be a non-empty set in ${\mathbb R}^n$ defined by a set of polynomial inequalities with rational coefficients $P_i(x_1, \ldots ,x_n) \gt 0 (1 \leq i\leq m)$ and $Q_j(x_1, \ldots ,x_n) \geq ...