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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Multivariable Optimization

I'm struggling to figure out how to go about this question: You are to produce a concrete box with an open top with a volume of 1$m^3$, having a wall and base thickness of $2$cm, by pouring concrete ...
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67 views

Arc length contest! Minimize the arc length of $f(x)$ when given 3 conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
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9 views

Using Lagrangian multipliers to check the solution

suppose we have an objective function $f(x)$ that we want to maximize subject to constraint $g(x)=c$. What I do next is set up Lagrangian optimization as follows and take the derivative ...
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1answer
16 views

how to find the solution of this cost function?

I have the following cost function. $J = \sum_{i=1}^N a\, Trace(W^TX_iW) - b\, Trace(W^TY_iW)$ Where $X_i$ and $Y_i$ are symmetric matrices, $a$ and $b$ are scalars. How can I find W?
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Linear optimization with constraint on the sum on “consecutive” indexes

I am struggling with the following optimization problem. Given $\{\lambda_i\}_{i=1\ldots k} > 0$ consider $\max_{x_i \in A} \sum_{i=1}^k \lambda_i x_i$ where $A = \{x_i \in \mathbb{R} : ...
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does a closed form solution exist for this equation?

I have a cost function $J$, which depends on a projection matrix $W$, which is unknown. When I get the partial derivative $\frac{\partial J}{\partial W}$ the equation is: $\frac{\partial J}{\partial ...
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1answer
29 views

Eigen value system? solution

I have the following system. $AW = \lambda B W$ Where $A,B,W$ are matrices and $\lambda$ is a scalar. The values of $A,B$ and $\lambda$ are known. $B$ is invertible. This is a solution to an ...
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4 views

Unique minimizer of $\|x\|_{\mathcal A}$ subject to $\Phi x=\Phi x_0$

I'm trying to understand the proof of Lemma 2.3 of the paper Simple bounds for recovering low-complexity models. The authors want to find bounds on the numbers of rows $m$ of $\Phi$ to ensure that ...
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3 views

How to know the rate of convergence of a majorization - minimization algorithm?

The basic idea of majorization-minimization (MM) principlein optimization is to convert a hard problem (for example, non-smooth) into a sequence of simpler ones (for example smooth). To minimize ...
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1answer
31 views

Founding maxima or minima to a function

$g(x)=e^{x-1}+x^{2}-3+2x$ How can I find when this function has maxima and minima? I found the derivative but I can't understand how find the solution when $g'(x)=0$. It's high school material.
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1answer
17 views

Network/graph theory -acyclic problem [on hold]

Consider an acyclic directed network of n vertices, labeled $i=1...n$, and suppose that the labels are assigned such that all edges run from vertices with higher labels to vertices with lower. Show ...
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1answer
30 views

Remove minimal number of elements

Given the numbers $ 1,2,..,2n + 1 $ , $ n > 0$ , remove as few numbers as possible so that among the remaining numbers no number is equal to the sum of two other numbers. After removal of first ...
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32 views

Maximizing sum of sign functions

I want to solve the following optimization problem: $$ \max_{{\bf u}} \sum_{i=1}^N \left(a_i \textrm{ sign}({\bf u \ . x_i}) \right), $$ where ${\bf u}$ and ${\bf x_i}$ are $p\times1$ vectors, $a_i ...
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26 views

Analyze the variation of this function $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$ w.r.t. $x$

Please, I need to analyse the variation of the following function w.r.t. $x$ : $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$, where $E_1[a+b (x-1)]$ is the exponential integral, $b>a$, $a>0$, ...
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1answer
26 views

Constraint minimization of sum of Non-symmetric matrices

I am trying to find closed form solution to following problem \begin{equation} \begin{array}{c} \text{min} \hspace{4mm} \big(\lambda_1\left( \mathbf{y}^T V^{(1)}\mathbf{x} \right)^2 + ...
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7 views

Assessing the “Quality ” of a solution sobtained by using lagrangian multipliers

I have an ill-defined question. I work in machine learning and am trying to learn the parameters of a model, such that my problem amounts to constrained optimization. That is, I have some training ...
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10 views

Local optimization with multiple starting values \approx global optimization?

I need to find the minimum/maximum of a nonlinear function but the constraints in the optimization problem make it tougher to solve (not a convex problem). I don't have a good global optimization ...
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39 views

How many additional crews should be brought in to minimize the cost of an oil spill?

An oil spill has fouled $200$ miles of Pacific shoreline. The oil company responsible has been given $14$ days to clean up, after which a fine will be $10000$ \$/day. The local cleanup crew cleans $5$ ...
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1answer
18 views

How do I set a lower bound to the solution's norm in a QP problem

I know that LASSO-regularization can be used to scale into an $L_1$ upper bound for a solution. But what if I want the norm to be within a specific range $[a,b]$? ie. I also want to set a lower bound? ...
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Wrong ILP solution with LPSolve (simple example)

I added the following example into LPSolve and found a strange issue. I don't want S1 and S2 to overlap within certain margins. ...
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1answer
18 views
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Reducing an I-optimal problem to a Pareto-optimal problem

Given a set $\textbf y\subset\mathbb R^2$, let $y = (y_1,y_2), y'=(y'_1,y'_2)\in\textbf y$ be elements of that set, let $\alpha_{min}\in\mathbb R$, $\alpha_{min}<1$, $\alpha_{max}\in\mathbb R$, ...
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Tangent cone to the graph and epigraph

Good morning! I am solving this example: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function given by $f(x) = \left\{ \begin{array}{rl} x \cdot sin(\frac{1}{x}) & \text{if } x > 0,\\ 0 ...
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1answer
12 views

Partition of fractional parts where each sum of them has to be at least 1

Let $ a_1,\ldots,a_t \in \mathbb{Q} \setminus \mathbb{Z} $ be with $ \sum_{i=1}^t \lbrace a_i \rbrace \in \left[k,k+1\right) $ for some $ k \in \mathbb{N} $ with $ k \ge 4 $. Here $ \lbrace x \rbrace ...
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Convexity of matrix inverse

If $$f\colon\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$$ where $$f(x,y)=y^T x^{-1}y$$ and dom$(f)=\{(x,y)\ |\ x+x^T\gt 0\}$, then is $f$ convex?
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15 views

Newton Raphson method to optimize two parameters using mathematica [on hold]

Am working on computational Thermal engg. work. For the optimization i have been using Mathematica Software. I got a expression "G" which I deduced in mathematica itself. But when I tried to find the ...
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Equivalence of the partial least square regresssion's iterative algorithm and its optimization problem

I am reading The Elements of Statistical Learning. This is a page from the partial least square section: The exercise asks to prove the equivalence between Algorithm 3.3 and Eq. (3.64). Here's my ...
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72 views

The area visible from two lighthouses with angle of vision 30 degrees, built at distance 10km from each other

The distance between 2 lighthouses is 10 km. What is the maximum area of the ocean in which both can be simultaneously visible if the angle of vision for each lighthouse is 30 degrees?But the minimum? ...
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28 views

Dimension of Polyhedra [on hold]

Can someone explain the question below? I'm pretty new in this area, and I did not understand anything. Question; Let $P$ be defined by the following $$\begin{align}x_1+x_2+x_3&\le ...
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1answer
65 views

Checking: finding extremals for a functional

I'm trying to find the extremals of the functional $$J[y] = \int_0^1 (y')^2 + y^2 + 4ye^x \, {\rm d}x,$$ imposed that $y(0) = 0$ and $y(1) = 1 $. I got that there can't be extremals, and that's weird ...
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1answer
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Polygonal chain in a rectangular parallelepiped

Given a rectangular parallelepiped ABCDA1B1C1D1 with edges AD = 6, AB = 8, AA1 = 8. Points M and N are the middles of A1B1 and C1D1. Points E and F are chosen on the edges CC1 and DD1 so that C1E = 3, ...
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116 views

Find $\int_0^a{f(x)}\, dx$

SMT 2013 Calculus #8: The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, ...
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Determining active constraints in KKT

Suppose there is a constrained optimization problem having inequality constraints. We can solve it using Karush-Kuhn-Tucker conditions. My question is how do we determine which constraints are active ...
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1answer
32 views

Is $f(x,y)=-xy$ neither concave nor convex?

Is $f(x,y)=-xy$ neither concave nor convex? I used the definition for first differentiable functions and determined it depends on the choice of points, hence it is neither.
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1answer
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Finding the lowest upper bound of product of two number using Young's inequality

Young's inequality for product can be stated as follows: $ab \leq \frac{1}{p}a^p + \frac{1}{q}b^q$ where a and b are nonnegative real numbers and p and q are positive real numbers such that 1/p + 1/q ...
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Gradient function for restricted likelihood with respect to terms that influence Sigma

Is there a straightforward/generalized way to calculate partial derivatives of the restricted multivariate log-likelihood function $\ln\mathscr{L}=C+\ln\lvert ...
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2answers
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Infimum of $\frac{||u'||^p_{L^p}}{||u||^p_{L^p}}$ for $u \in W^{1,p}_0((0,1))$

Good afternoon everyone! It is very easy to show that the infimum mentioned in the title is strictly positive, but it seems much more difficult to show that it is attained within the Sobolev space of ...
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refomulation of an optimization problem

I have written a program for optimizing a set of generators. And I need to reformulate this problem, to include additional generators and constraints. I have hourly price and cost data and need to ...
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1answer
27 views

Maximal Multiplication of All Possible Summands

I have recently got interested in the following problem: Give a decomposition of a natural number to natural summands whose multiplication is maximal. I have tried to solve this problem, and ...
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second derivative fail, classify the nature of the critical point

f(x,y,z)= $\frac{(x+y)^2}{2}+z^3$ the critical point I calculated is span{(1,-1,0)} the eigenvalue of the Hessian of point (1,-1,0) are 0,0,2, which means that this point is degenerate and the ...
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1answer
16 views

Equality Constraints in Quadratic Programming

Now I am new to the world of primal-dual algorithms and I want to understand the SOCP-Code of Lobo/Vandenberghe/Boyd (primal dual interior point method). Currently I am working through Goldfarb and ...
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1answer
30 views

How to show $ \sup \inf g(x,y) \leq \inf \sup g(x,y)$?

Came across this little practice exercise, and I couldn't properly convince myself of this relation: Let $X,Y \subset \mathbb{R}^n$ and $g:X\times Y \rightarrow \mathbb{R}$. Show that $$\sup_{y \in ...
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How do i optimize a function with an integral in it?

I want to find the values of $t_1 t_2...t_n$ which would give the maximum value for the function below: $\int_a^b \{f_1(x-t_1)+f_1(x-t_2)+f_1(x-t_n)\} dx$ functions $f_1f_2....f_n$ are expected to ...
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Positive/Negative Definite Bordered Hessian?

I understand how to check a function for concavity and convexity using the Hessian matrix and the rules for the determinants of the leading principal minors. I understand if these rules are violated, ...
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26 views

where can I find Normal Boundary Intersection method matlab codes? [on hold]

I have a constraint optimization problem. I want to convert it to a multi-objective mathematical programming problem (Multi-objective optimization , vector optimization). I need to run Normal Boundary ...
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15 views

maintaining monotonicity in an optimization problem

I have written a program for optimizing a set of generators. I have hourly price and cost data and need to figure out when a generator should run or just stay off. I describe the problem in more ...
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1answer
24 views

Biobjective optimisation, pareto non-domination

Ok, so, I have a function $f_I(y_1, y_2) = \max\{\alpha y_1 + (1-\alpha)y_2:\alpha\in[\alpha_{min},\alpha_{max}]\}$ that I'm trying to minimise, and I'm asked to find, amongst a set of vectors $y$, ...
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1answer
17 views

Problems with vector vector derivative in optimization

I have a loss function of the followoing form: $L(\mathbf{a}) = \|\mathbf{b} - \mathbf{a}\|_2^2$ Where, $\mathbf{a}$ and $\mathbf{b}$ are vectors of dimension $d\times 1$. I need to calculate ...
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26 views

Maximin problem as LP?

Consider the following setting. Let $A\in \mathbb{R}^{3 \times m}$ and $B\in \mathbb{R}^{m\times 3}$ be two matrices such that each of their columns must add up to a given $c\in \mathbb{R}$. Denote by ...
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1answer
57 views

How to minimize the number of functions to be projected on

I have a set of functions $f_i,\, i=1,2,\ldots,n$ defined on an interval $[a,b]$, and a function $F$ also defined on $[a,b]$. I would like to project F on a subset of functions $f_i$ so that the ...
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1answer
15 views

Compute the generalization of average

Consider $n$ real numbers $a_1, \ldots, a_n$. Let $x_p = argmin_{x} \sum_{i=1}^n |a_i-x|^p$. I know $x_2$ is the average. $n$ is odd, $x_1$ is the median. How about other $p$'s. Are there simple ...