matrix – Doubt about notation in Robust Optimization

I’m studying the pricing model under Robust Programming here described from page 41 to page 45. I don’t understand what’s the meaning of subscripts $i$ and “second” $t$ referred to dual variables of model. As a comparison, take the model at page 38: dual variable $p_{1,t}$ (included in the second constraint of case I) refers to the set $begin{Bmatrix}
p_{c,t}
end{Bmatrix}_{c=1}^4,_{t=1}^T$
, where $c$ indicates the constraint subject to conversion as part of reference set of four constraints and $t$ the subperiod to which is associated the single realization of returns. Matrix disequality of page 37 clears it up. Instead, in the model that I’m studying now, the reference set becomes $begin{Bmatrix}
p_{c,t,i}
end{Bmatrix}_{c=1}^4,_{t=1}^T,_{i=1}^T$
.

So:

  1. What does $i$ say?

  2. Why, for example in case t.I third constraint, do we have $p_{t,1,t}$? What does the “second” $t$ mean?

Thanks in advance for any help.

optimization – Separating labelled points with a plane, minimizing label variance

Suppose we have observations with associated labels ${({bf x}_1, y_1), ({bf x}_2, y_2), dots, ({bf x}_n, y_n)}$ where ${bf x}_i in mathbb{R}^d$ and $y_i in mathbb{R}$.

Can we efficiently find a separating plane of the observation space $mathbb{R}^d$ such that the sum of the variances of the labels of the observations within both groups is minimized?

That is, if a separating plane is parametrized by $({bf v}, c)$ splitting the space into groups ${bf x}_i^T {bf v} < c$ and ${bf x}_i^T {bf v} geq c$ we can formalize our search problem as:

$$arg min_{{bf v}, c} {Big (text{var}({y_i : {bf x}_i^T {bf v} < c}) + text{var}({y_i : {bf x}_i^T {bf v} geq c})Big)}$$
where $text{var}(S) = frac{1}{|S|}sum_{tin S}{big(t – frac{1}{|S|}smallsum Sbig)}^2$ is the standard definition of variance, optionally with Bessel’s correction applied (either definition is fine).


The context is that I had the idea of replacing the simple single-attribute decision nodes of decision trees with separating planes instead. The above method would find the greedy best split. I was wondering if doing this is even tractable?


Note that given $bf v$ we can find optimal $c$ in $O(n log n)$ time by computing all breakpoints ${bf x}_i^T{bf v}$, sorting, and computing the variances of each breakpoint in an online manner, choosing the best one as $c$. This gives us a randomized approximation algorithm, by repeatedly choosing a random unit vector $bf v$ and finding optimal $c$. This continues for as long as desired, and the best result is returned. In theory given enough time this should return the optimal result (since a minimizing $bf v$ is not unique and has constant probability of being chosen in any iteration), however I don’t know how well it can be expected to approximate given a certain number of iterations.

optimization – Alternative form of Gradient Descent

We define the standard gradient descent as follows:

$$x^{(k+1)} = argmin_{xinmathbb{R}^n} f(x^{(k)}) + langle nabla f(x^{(k)}), x – x^{(k)}rangle + frac{1}{2alpha_k} ||x – x^{(k)}||_2^2$$

Here, $x^{(k)}$ is the result of the $k$-th iteration of the gradient descent. Solving the above equation gives us $x^{(k+1)} = x^{(k)} – alpha_k nabla f(x^{(k)})$, which is our usual gradient descent.

How do we prove that the gradient descent problem is equivalent to the following version:

$$x^{(k+1)} = argmin_{xinmathbb{R}^n}langle nabla f(x^{(k)}), x – x^{(k)}rangle $$ subject to $$||x – x^{(k)}||_2 leq alpha_k||nabla f(x^{(k)})||_2$$

mathematical optimization – How to reverse or restore the order of my cells in Wolfram Mathematica Cloud?

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nonlinear optimization – Improving convergence of the Newton-Raphson method

How can the Newton-Raphson method (that is, the multivariate generalization of Newton’s method, used in the solution of nonlinear systems) be improved so as to attain better convergence? As-is, in most cases a fairly good initial value is required to ensure convergence.

Note that I’m aware other method exist. Here, I’m simply interested in alterations/modifications to Newton’s method.

mathematical optimization – NMinimize to ignore arguments for which function is complex valued

I’d like to minimize a pretty complicated function $f(x, y, z)$ with NMinimize, however for certain values of the arguments $f$ becomes complex valued, and my NMinimize essentially fails… Is there a way to make Mathematica only consider arguments for which the function I’m trying to minimize f is real-valued?

sql server – Long Running Query Optimization

I need to optimize this query as this query is taking minutes of time in execution.
Below is the query. i’m giving date parameters and shift id is optional.
as query is returning same in and out from attendance table based on shift ID.

“SELECT E.EmpCode,E.EmpName,e.DesignationName,D.DeptName, CONVERT(Date, A.TimeIn) AS Date, a.TimeIn,a.TimeOut, DATEPART(WEEKDAY, CONVERT(date, A.TimeIn)) AS Day,sh.ShiftID,CONVERT(TIME,sh.TimeIn) AS STimeIN,CONVERT(TIME,sh.TimeOut) AS STimeOut,sh.ShiftName,
CASE WHEN CONVERT(TIME,a.TimeIn) < CONVERT(TIME,DATEADD(HOUR,4,sh.TimeIn)) THEN ‘MarkOut’ ELSE ‘MarkIN’ END AS Status
FROM dbo.Attendance AS A
INNER JOIN dbo.Employee E ON A.EmpID = E.EmpID
INNER JOIN dbo.Department D ON E.DeptID = d.DeptID

                                     INNER JOIN dbo.Shifts sh ON sh.ShiftID = (SELECT s.ShiftID FROM dbo.Shifts s , employee emp WHERE s.ShiftID = isnull((select ShiftID from EmpWiseShiftAllot where CONVERT(date,DateFrom) = CONVERT(date, A.TimeIn) and IsActive =1 and IsCancel =0 and empid = E.empid), 
                                    (select top 1 ShiftID from DeptStrengthLimit L where L.DeptID = E.DeptID and L.CompanyID = 2 AND L.IsCurrent = 1            
                                    AND CONVERT(date, A.TimeIn)  between L.WithEffectFrom and CONVERT(date, A.TimeIn) order by L.shiftid desc)) and empid = E.empid )

                                     WHERE (A.TimeIn BETWEEN '2020-11-17'  AND '2020-11-21' ) AND (DATEDIFF(MINUTE, A.TimeIn, A.TimeOut) = 0) 
                                         AND 
                                        (A.EmpID NOT IN (SELECT M.EmpID FROM dbo.LeaveApplicationD AS D INNER JOIN  dbo.LeaveApplication AS M ON D.LeaveMID = M.EmpLeaveID
                                          WHERE        (D.IsActive = 1) AND (D.IsCancel = 0) AND (M.IsActive = 1) AND (M.IsCancel = 0) AND (D.Date = CONVERT(date, A.TimeIn) ))) 
                                          AND (DATEPART(WEEKDAY, CONVERT(date, A.TimeIn)) <> 1)
                                                    AND (E.CompanyID = 2) AND (E.IsActive = 1) AND (E.IsCancel = 0) 
                                              AND sh.ShiftID = 5 ORDER BY date"

optimization – Proof of Max-Min Inequality

Is this a correct proof of the Max-Min Inequality? And if not how could it be improved? Thank you for your help.

Given $gcolon Xtimes Yto mathbb{R}$, we note that for any $tilde{x} in X$, $tilde{y}in Y$ that

$$inf_{xin X}g(x,y) leq g(tilde{x},tilde{y}) leq sup_{yin Y}g(x,y)$$
so
$$inf_{xin X}g(x,y) leq sup_{yin Y}g(x,y)$$

Taking the supremum over all $yin Y$ leaves the right-hand side unaffected so
$$sup_{yin Y}left(inf_{xin X}g(x,y)right)leq sup_{yin Y}g(x,y)$$
Finally take the infimum over all $xin X$ yields the desired result
$$sup_{yin Y}left(inf_{xin X}g(x,y)right)leq inf_{xin X}left(sup_{yin Y}g(x,y)right)$$

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oc.optimization and control – how to draw a diagram of a rectangle using optimization

the question is
A rectangle has its two lower corners on the x-axis and its two upper corners on the curve y=10e^-(x^2)/18.

A) draw the diagram

I am not being lazy and asking someone to just solve it. I genuinely need to learn how to solve and would like to compare the correct answer with mine