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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the implication of zero mixed partial derivatives for multivariate function's minimization

Suppose $f(\textbf x)=f(x_1,x_2) $ has mixed partial derivatives $f''_{12}=f''_{21}=0$, so can I say: there exist $f_1(x_1)$ and $f_2(x_2)$ such that $\min_{\textbf x} f(\textbf x)\equiv ...
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Maximize area of a triangle with fixed perimeter

If perimeter of a triangle is $2d$, what is the length of sides so the triangle has maximal area? I found some solution using circle and angles, but I think I have to use derivatives. I need ...
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130 views

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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214 views

Are these convex optimization problems equivalent?

Consider the optimization problem $$ \mathcal{P}_1: \qquad \min_{x \in \mathbb{R}^n} c^\top x \quad \text{sub. to } \ g(x,y_i) \leq 0 \ \ \forall i = 1,2,...,M$$ where $c \in \mathbb{R}^n$, and ...
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322 views

Showing a dual LP solves a primal LP

I originally asked this question: Does solving the LP dual SOLVE the primal LP? It was answered using an example of how the primal and dual solve each other (because of knowledge from strong ...
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730 views

A constrained linear least Frobenius norm problem:$\min_{X} \|A-XB\|_F$ subject to $Xv=0$?

Assume we are given two matrices $A, B \in \mathbb R^{n \times m}$ and a vector $v \in \mathbb R^n$. $\|\cdot\|_F$ is the Frobenius norm of a matrix. How can we solve $$\min_{X \in \mathbb R^{n ...
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550 views

How would you solve a Tikhonov Regularized Least Squares problem with nonnegative constraints?

For a Tikhonov Regularized Least Squares problem with nonnegative constraints, what are some methods that solve it? Are methods solving a Least Squares problem with nonnegative constraints and the ...
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79 views

Is $ \sum_{1 \le k \le n} (y_k - a x_k^b + c x_k^d + e)^2 $ convex?

Over at How many points to find a polynomial? it was suggested to minimize $$ f(a,b,c,d,e) = \sum_{1 \le k \le n} (y_k - a x_k^b + c x_k^d + e)^2 .$$ However I don't know if it is possible to find ...
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Max distance between a line and a parabola

I am suppose to use calculus to find the max vertical distance between the line $y = x + 2$ and the parabola $y = x^2$ on the interval $x$ greater then or equal to $-1$ and less then or equal to $2$. ...
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521 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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481 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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85 views

Minimial Tiling Problem on a sphere

This question is a revision of the math exchange post found here. Consider the following: A sphere, $S$, with radius $r_1$. N regions projected onto $S$, whose projections, $\left\lbrace E_i ...
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621 views

Sign restriction on the Lagrange multiplier? Why?

Say we are given a linear program where the goal is to minimize $c^Tx$ with the constraints $Ax\ge b$. Why is there a sign restriction on the Lagrange multiplier associated with the active constraints ...
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Set boundary for least-square calculation

When calculating least-squares we use the form $$Xw=y$$ This gives us an approximation, and it can happen that the resulting $w$ will be such that $Xw \lt y$ (entry-wise). Is there a way how to ...
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70 views

Upper bound of an expressions with many variables

Assume $0 < p_1 \le p_2 \le \dots \le p_{2k}$. I am looking for a (preferably tight) upper bound for the following expression: $$ \frac ...
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323 views

Underlying assumption in a Primal/Dual table

I just read in one of the questions answered by @MikeSpivey that the following table is provided in Sierksma's Linear and Integer Programming: Theory and Practice, Volume 1, page 144. ...
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334 views

Sudoku mathematically, MILP?

My homework contains a word (freely-translated) "target-function" that I should generate somehow for 9x9 sudoku solver with some MILP problem. But I am bit lost what they mean. I have sofar described ...
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809 views

Optimizing with Absolute Value Objective Function

max : $w = |q^T y|$ subject to $A y \leq b$ $y \geq 0$ Please describe how one could solve the non-linear programming prob. above by using linear programming methods. I tried changing $y$ to $y' ...
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215 views

Voronoi diagram with different metric functions

Given a metric space $(X,d)$ and finite number of points $(x_i)_{i=1}^n$ the Voronoi diagram (or the Dirichlet cell) $C_i$ is given by $$ C_i = \{x\in X:d(x,x_i)<\min\limits_{j\neq i}d(x,x_j)\}. ...
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248 views

Questions about weak duality theorem

Following are some corollaries regarding the weak duality theorem. Consider a constrained problem, $\min_{x \in X} f(x),$ subject to $g(x) \leq 0$ and $h(x) =0$. Its dual problem is $\sup_{u \geq ...
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120 views

Minimum for this function

I thought of writing this question Minimum for this function in a different way, if it helps. I want to minimize $$\sum_{i=1}^n a_ix_i + \nu \sum_{i=1}^n b_i 2^{x_i} ,$$ where $a_i \in [0,1]$, ...
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158 views

Find the range of $x$, given $y_{min} \leq y(x) \leq y_{max}$, where $y(x) $ can be any function ( Updated)

I have a series of inequalities: $$y_{1min} \leq y_{1}(x) \leq y_{1max}$$ $$y_{2min} \leq y_{2}(x) \leq y_{2max}$$ $$..$$ $$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$ Note that $x\in\mathbb{R}$ The ...
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The solution of $\min_x \max_{1\leq r \leq N} \left|\frac{\sin rMx}{\sin rx}\right|$

I am looking for the solution of $$\min_x \max_{1\leq r \leq N} \left|\frac{\sin rMx}{\sin rx}\right|$$ where $M < N$ are integers and $x \in \mathbb{R}^+$. For $M = 4, N = 6$, $f_{r,M}(x) ...
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What's wrong in this dual derivation?

I have a function in the form \begin{align} f(q,M)=\sup_{0\leq \alpha \leq 1} -\alpha^T (R\odot M)\alpha+\alpha^Tq \end{align} which is a dual of a minimization problem, where $R$ and $M$ are ...
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41 views

Minimization involving equality constraints

I am trying to find closed form solution to following problem \begin{equation} \begin{array}{c} \underset{\mathbf{x},\mathbf{y}}{\text{minimize}} \hspace{4mm} \big(\left( \mathbf{y}^T ...
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58 views

Definition issue with limiting directions

In Nocedal/Wright's Numerical Optimization (1999) in section 12.3 the notion of feasible sequences and related limiting directions are introduced as a starting point for the proof of the ...
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26 views

The functional take its maximal value for $y(t)=-t$

I want to show that the functional $J(y)=\int_0^1 [y'(t) \sin{(\pi y(t))-(t+y(t))^2}]dt$ ,where $y$ is a continuously differentiable function on $[0,1]$, takes its maximal value $\frac{2}{\pi}$ for ...
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How can we maximize the following functional?

$\max_{} \; \int_0^1 \left( -\frac{1}{2} \left( \lambda_1(1-t) - \int_t^1 \lambda_2(s) ds \right)^2 - 1.25 \lambda_2(t) \right)dt + \lambda_1$ s.t $\lambda_1\geq0$, and $\lambda_2(t) \geq 0$ for ...
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Do three valued basis vector elements lead to the fastest discrete Fourier transforms?

When sin() and cos() are approximated to 1, 0 and -1 in the basis vectors in a real or discrete Fourier transform the basis vectors have a lot of elements of zero or in common leading to an algorithm ...
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65 views

Difference between mean-variance and worst-case optimization for normal distribution

I have two optimization problems. 1-) Mean-variance optimization $J_{MV} = J_M - \gamma J_V$, where $J_M$ is mean, $J_V$ is the variance term and $\gamma$ is the weight on variance term. 2-) The ...
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How to determine the gradient of this cost? [duplicate]

I have asked a similar question before, but I guess I haven't provided clear information. The cost of my function $f:\mathbb{R}^5\rightarrow\mathbb{R}$ is $$f(\vec{\alpha}) = ...
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Problem about length and width of a running facility

Jacaranda Secondary College is planning to develop a $400$ metre running track facility in an unused area of the college. The rectangular site available is $100$ metres wide and $180$ metres long. The ...
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Proof that nuclear norm is convex.

For an $m \times n$ matrix, $A$, the nuclear norm of $A$ is defined as $\sum_{i}\sigma_{i}(A)$ where $\sigma_{i}(A)$ is the $i^{th}$ singular value of $A$. I've read that the nuclear norm is convex ...
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Find $\min x^TAy+b^Tx+c^Ty$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$

The problem seems to be easy but I can't find a solution :( Problem: Given $A\in\mathbb{R}^{m\times n}, A\ge 0, b\in\mathbb{R}^{m}, c\in\mathbb{R}^{n}$. Minimize $f(x,y) = x^TAy+b^Tx+c^Ty$ subject to ...
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137 views

An Interesting Resource Allocation Problem

Here is the problem: \begin{array}{ll} \text{minimize} & \sum_{i=1}^N \frac{1}{1 + \textrm{exp}(C_i + x_i)}\\ \text{subject to} & \sum_{i=1}^N x_i \le R \\ & x_i \ge 0, ~ i = 1,2,...,N ...
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39 views

Looking for Method to evaluate the optimal node rate vs number of simulation rate in a Monte Carlo simulation

I am currently working on evaluating an American Option using a Monte Carlo simulation, and I am getting answers but they vary quite a bit. The two variables that I can alter are number of simulations ...
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68 views

Optimization to minimize cost using the function C=Tq^(1/a)+F

I was given the function of $C=Tq^{1/a}+F$ where $C$ is total cost, $q$ is output, $a$ is a positive parametric constant, $F$ is the fixed cost, and $T$ measures the technology available (also a ...
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1answer
26 views

Finding the optimal solution

Find the optimal solution of the problem $$\min \Bigg \{ \int_0^1 [x^\prime (t)^2 + 2x(t)^2]e^t dt : x(0) = 0, x(1) = e - e^{-2}\Bigg \}$$ and the value of the minimum. Not sure how to approach this ...
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Critical points characterization of real function

Providing a real multi-variable function $f(\bar{x})$ twice differentiable with respect of all its variables. Looking for critical points is equivalent to solve $\nabla f = \vec{0}$. And to ...
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Using bordered Hessian matrix to determine non-degeneracy and type of constrained extremum

I have the following problem: $\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}\def\g{g(x_1,x_2,x_3)}\def\l{\lambda}\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}$ Find the ...
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Optimize volume of an open cardboard box made from flat square of cardboard…

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. ...
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monotonicity of binomial coefficient

I am interested in $$f(x):={k-1 \choose x-1} p^{x} (1-p)^{k-x}.$$ How do I find out in which Domain this function is monotonically increasing, in which it is monotonically decreasing? For which $x$ ...
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Maximum payoff for safe bet

I'm having a hard time choosing a good strategy for this problem: assume that you have $m$ money that you can bet on $n$ mutually exclusive outcomes, all with unknown probabilities, and that each ...
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129 views

Maximizing the volume of a box using Lagrange multipliers

We are given a box of surface area $64$. As such, I wish to maximize $f(x,y,z) = xyz$ subject to $g(x,y,z) = 2(xy+xz+yz) - 64$. If I have understood in correctly, I am to find the critical points of ...
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Are there known patterns among minimal expressions?

Let $R = F[z_1, z_2, \dots]$ be the finite-degree polynomials in a countable number of variables. Let $\mathcal{E}(R)$ be the set of all expressions of polynomials. Note that there could be an ...
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Minimize Function over Convex Subset

Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ...
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Simple Lagrange Multiplier Problem, not working out

The question should be simple. Use the Lagrange Multiplier to maximize $f(x,y) = 4x^2 + 10y^2$ subject to the constraint $x^2 + y^2 = 4$. But when I set it up I get two different values for ...
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54 views

Evaluate smartly a function on a multiplication grid

I am asking myself the following question: Suppose one has a grid $G \in \mathbb{N}^{n\times n}$ where $g_{ij} = i\cdot j$, $i,j \leq n$. I would like to evaluate a function $f: G \to \mathbb{N}$. ...
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168 views

Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
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831 views

Calculus Optimization - Finding the minimum cost

In oil pipeline construction, the cost of pipe to go underwater is 60% more than the cost of pipe used in dry-land situations. A pipeline comes to a river that is 1 km wide at point A and must be ...