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

Contest: Give an example(s) 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 $f$ ...
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1answer
19 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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369 views

Convex Optimization of quadratic function with inequality constraints

How would I solve the following problem? $$\min_{x\in\mathbb{R}^n} x^T A x$$ subject to the constraints $$x_i\geq 1,\,i=1,\dots,n,$$ where A is positive semidefinite and symmetric. Is it possible to ...
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1answer
12 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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20 views

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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33 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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26 views

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
18 views
+50

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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0answers
11 views

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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8 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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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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0answers
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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1answer
32 views

Understanding relation between vector valued function and function objective in an multi objective optimization problem

I try to understand the relation between "vector-valued function" and "function objective" as used in optimization problem. I understand that objective function in a multi-objective problem can be ...
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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
27 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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0answers
88 views

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 ...
2
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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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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
19 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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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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2answers
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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41 views

Understanding a quadratic optimization problem

I'm relatively new to this kind of problem since I never faced a quadratic optimization problem before. The following question builds on a previously question of mine which has been kindly answered ...
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1answer
269 views

Absolute extrema

Find the absolute extrema of the function $f(x,y)=2xy-x-y$ over the region of the xy-plane bounded by the parabola $y=x^2$ and the line $y=4$ I was wondering if I needed to use Lagrange multipliers to ...
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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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11 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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1answer
40 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
53 views

Was my inference regarding $u(z)=log(z)$ correct?

I have already solved the problem but would appreciate a clarification in part (b). A has initial wealth $w$ and faces a loss $l$ with known probability $\pi$. Insurance available at unit price ...
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29 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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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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10 views

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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22 views

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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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 ...
3
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2answers
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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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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1answer
12 views

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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268 views

Optimization, rectangle inscribed inside arch of the curve.

A rectangle is to be inscribed under the arch of the curve $y = 4\cos(0.5x)$ from $x = \pi$ to $x = -\pi$. What are the dimensions of the rectangle with largest area, and what is the largest area? ...
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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
27 views

strong convexity of loss function in multi-dimensional (high-dimensional) space

My question is based on this paper (see the last 10 rows in page 7). It seems this is a general claim: In machine learning or statistic, the loss function $l(W^TX, y)$ (a linear predictor) can never ...
2
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0answers
11 views

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.
3
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1answer
39 views

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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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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116 views

Can gradient descent solve this problem $\text{argmin}_x \|Ax-[Var(Ax)]^{\frac{1}{2}}-b\|^2$?

How can I find the (approximate) solution to the following problem: $$\text{argmin}_x \|Ax-[Var(Ax)]^{\frac{1}{2}}-b\|^2,$$ where $Var(.)$ denotes the variance? $A$ is matrix and $b$ and $x$ are ...
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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 $y$, ...
0
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2answers
543 views

How to calculate the hessian of a matrix function

I am estimating a model minimizing the following objective function, $ M(\theta) = (Z'G(\theta))'W(Z'G(\theta))$ $Z$ is an $N \times L$ matrix of data, and $W$ is an $L\times L$ weight matrix, ...
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31 views

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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16 views

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 ...