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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Lattice fitting to points

I have a set of points (shown as little black circles) which ideally form a hexagonal lattice shape, each point having an equal distance to all of its neighboring points. (Sorry for my drawing, some ...
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If a continuous function has one relative extremum, then it is actually an absolute extremum

The following statement makes sense intuitively, but is there a way to prove it mathematically? (This is something we make use of in applied optimization in calculus.) If $f$ is continuous on an ...
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maximization of a particular ratio

We are given a ratio: $$\frac{g(x)}{f(x)}$$ where: $$g(x) \in \mathbb{R}^{+}$$ $$f(x) \in \mathbb{N}\: \cap f(x)\ge 2$$ So $g(x)$ returns values in $[0,+\infty]$ while $f(x)$ returns values in ...
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Finding critical points of a multivariable function

Let $f(x,y)=e^{x^2-xy+y^2}$ (a) Find all the critical points of the following function. (b) Find the all the local maxima and local minima of the function if there is any. What i tried. I tried ...
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Optimization Question Regarding proving Maximum Area of Window

A window has the shape of a rectangle of height $h$ surmounted by a semi-circle of radius $r$. The area of the window is given by $A = 2rh + \frac{1}{2}\pi r^{2}$ NOT drawn to scale. If the ...
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79 views

maximize function with two variables

I would like to maximize the function: $f(x,y) = c[x\log(2y) + (1-x)\log(2(1-y))]$ subject to constraint that $x,y \in (0,1)$ to find a relationship between $x$ and $y$ that maximizes $f(x,y)$. My ...
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Formulating Solution for Branch and Bound

I have a linear programming question which I am setting up for a branch and bound solution. I am having issues with where to begin. The question is asking to find the minimum operating cost to ...
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86 views

Boundary solution to a max-min problem

Find the maximum and minimum values of $$f (x, y) = x − x^2 + y^2$$ on the rectangle $0 ≤ x ≤ 3, 0 ≤ y ≤ 2.$ I derived the determinant of the corresponding Hessian matrix and it turn out to be ...
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Find max/min values of the sum of squares

How to find max/min values for the sum of squares: $n_1^2 + n_2^2 + ... + n_i^2$ where $n_1 + n_2 + ... + n_i = c$ Is it true that max value is always obtained when $n_1 = n_2 = ... = n_i$?
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Optimization Calculus Question

Find the maximum point on the graph $$y = x^a(1-x)^b$$ where $a > b$ and the interval of $x$ is $0 < x < 1$.
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Global maxima/minima of $f(x,y,z) = x+y+z$ in $A$

Find the global maxima/minima of $f(x,y,z) = x+y+z$ for points inside of $A = \{ (x,y,z) \in \mathbb{R}^3: x^2-y^2 = 1 \wedge 2x+z = 1 \}$ I renamed the conditions of $A$ to a function $g(x,y,z) ...
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Finding Optimization of Rectangle

I have the following problem and I want to find the minimum. A rectangle fence is being built. One side of the rectangle costs 5 dollars per foot, while the other 3 sides cost 3 dollars per foot. The ...
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116 views

Linear independence of equality constraint gradients in constraint qualifications

I'm, trying to get an intuitive feel for the various constraint qualifications for KKT points. Most of them seem to rely on the linear independence of $\nabla g_i(x^*)$ where $g_i$ are the equality ...
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L(x)<=U(x) & L'(x) and U'(x) exist. Assume there's a pt c st. U(c)=L(c). Consider U(x)-L(x) and show that c is a min of this function

Question and attempt at question are in the photo below. I have gotten halfway through but I am confused how to show the rest of the question (mainly part c) Thanks for your help in advance :)
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61 views

Converting a Summation to an integral

Please how do I convert this summation $$ \frac{r-1}{n} \sum_{i=r}^n \frac{1}{i-1} $$ to the integral $$ x \int_x^1 \frac{1}{t} dt = -x \ln x? $$ by substituting $x = r/n$, $t=1/n$ and $dt =1/n$. ...
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Optimization with changing objective function

Is there any theory about (convex) optimization where the objective function is allowed to change during the optimization process? I have a problem where the objective function depends on some ...
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Why does gradient descent make sense?

Suppose I define two functions of $x$ in terms of a convex function $f$ with a unique minimum $x_0$: $$f_1(x) = 1 \times f(x)$$ $$f_2(x) = 2 \times f(x)$$ Suppose I wanted to minimize each of these ...
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141 views

A problem on positive semi-definite quadratic forms/matrices

Suppose $a+b+c=0$ and (without loss of generality) $a\leq b\leq 0\leq c$, $a^2+b^2+c^2=1$, is the following quadratic form positive semi-definite? Thank you very much. \begin{equation*} \begin{split} ...
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64 views

Taking the dual of this non-standard linear program

I am just beginning to learn linear programming have a question about taking the dual of a non-standard LP specifically the one below: $\min M\\ 2x_1 + 3x_2 + 4x_3 \leq M \\ 2x_1 - x_2 + x_3 \leq M\\ ...
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367 views

Convergence of Steepest Descent: Proving Orthogonality of Exact Line Search Steps

For the following assume that $f(x) = 0.5x^TQx - b^Tx$, where Q is symmetric, positive definite $n$ x $n$ matrix, and $b$ belong to $R^n$. Assume that $x^*$ is the unique local minimizer of $f(x)$ and ...
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102 views

Meal Platters Optimization Problem

Mark has to buy hamburgers, hot dogs, and pig's feet for an event. The restaurant he is purchasing from offers two Platter options. Platter A comes with 4 hamburgers, 3 hot dogs, and 2 pig's feet. ...
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Reference request: Time-optimal trajectories

I am looking for some lecture notes or a textbook for time-optimal trajectories. Any help is greatly appreciated. I am having plenty of trouble with understanding switching $C^+$ and $C^-$ curves.
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Help finding value of N that minimizes a sum

Suppose we have the following inequality: $\sum\limits_{k=N+1}^{1000}\binom{1000}{k}(\frac{1}{2})^{k}(\frac{1}{2})^{1000-k} = \frac{1}{2^{1000}}\sum\limits_{k=N+1}^{1000}\binom{1000}{k} < ...
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Second order derivation optimization

Recently I am thinking about a problem that might be easy to answer but for me is a big challenge. Assume you have a function $f(x)$ that is second order derivative. So I am looking for a way to ...
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272 views

Find a Square from n given Points.

Given set X of m Integeral Cords. I need to add minimum number of points to set X such that i get atleast one Square. For example: lets say X:{(0,0),(2,2),(3,3)}. Now i will have to add minimum 2 ...
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By minimizing the function $\phi(s,t) = \frac{1}{2} \mid\mid \textbf{b} - (s\textbf{a}_1 + t\textbf{a}_2) \mid\mid^2$, find a for

Suppose $\textbf{a}_1$ and $\textbf{a}_2$ are linearly independent vectors, $L = \text{span} \ \{{\textbf{a}_1, \textbf{a}_2}\}$, and $\textbf{b}$ is a vector not in $L$. By minimizing the function ...
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206 views

Minimize and maximize the function $g(a,b) = a + b$ given the constraint $h(a,b) = \frac{1}{a} + \frac{1}{b} = 1$.

I want to minimize and maximize the function $g(a,b) = a + b$ given the constraint $h(a,b) = \frac{1}{a} + \frac{1}{b} = 1$, and I want to find the values of $a, b,$ and $\lambda$. This is what I've ...
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Finding minimum on graph for given domain

So I want to find what is the minimum value of a graph on a certain domain. For example, for $y=x^2+1$ between $x=-3$ and $2$, the minimum value is at 1 at x=0. I think I know how to find minimums ...
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How to approach a minmax problem?

Starting with a certain geometric problem, I have reached this function: $$R(s,t,u,v)=\max(s-u,s+u,t-v,t+v,sX+tY+u, tX-sY+v)$$ where $X\geq0$ and $Y\geq0$ are parameters. I have to find the minimum ...
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Designing linear systems to respond to particular kinds of oscillations

Say that I have a linear system which is being perturbed by an oscillating signal of a single frequency, of the form $$ \dot{\vec{x}}(t) = A\vec{x} + B \sin(\alpha t), $$ where $B$ is a vector of 1s ...
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Properties determining boundedness of function

The function I am looking at is $$f(x) = \frac{1}{2}x^TAx + b^Tx + c$$ where $A$ is a symmetric matrix in $\mathbb{R}^{n\times n}$ and $b,c$ belong to $\mathbb{R}^n$ I want to determine what ...
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Coercive or not?

I had this problem in the exam. Let $X = [x_1,...,x_d]^T$, $a\space \epsilon$ $\mathbb{R}^d$ and $C$$\epsilon$$\mathbb{R}$. Argue for or against. $f(X) = a^TX + C||X||^2$ is coercive only for $C ...
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minimum value of $x^2+y^2+z^2$ subject to $ax+by+cz=1$

If $ax+by+cz=1$, what is the minimum value of $x^2+y^2+z^2$ It is obvious that we can do Lagrangian multiplier,$W=x^2+y^2+z^2-\lambda (ax+by+cz-1)$
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Number of Integer solutions for this optimization problem

What is the number of integer solutions to the problem $$\sum_{i=1}^{i=k}x_i = n$$ subject to $\forall_i\ \ x_i \ge 0 $ note This should hold for both cases $k < n$ and $k \ge n$
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How to solve this optimization problem with equality constraints?

I want to find $\delta_j$ in the following optimization problem. My variables are $\gamma_i$ and $\delta_j$ (all other symbols are known parameters). Assume $i\in\{1,\ldots,9\}$ and ...
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How to find local maximum of the function $f(x) = x^3-9x^2+24x+4$?

Give the value of x where the function $f(x) = x^3-9x^2+24x+4$ has a local maximum. a) -4 b) 4 c) 2 d) 3 e) -2 I graphed it and I'm not sure how to find the local max
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Teacher/student exam assignment matching problem - equivalent problem?

I have a sort of matching problem. I am wondering if you know if this problem reduces to a familiar one. It arises from my friend's job, and something we were wondering about this morning on the ...
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Does existence of global minimum imply coercivity?

It is known that a coercive function over a closed, unbounded set has a global minimum. Is the converse true ? The larger context for this question is the following question: Suppose we are given a ...
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Local minimum of the function:

Find the local minimum of the function: $$\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}$$ $$\f=\1^2-2\1\2+2\2^2+\3^2 \text{ in } \mathbb{R}^3$$ $\n\f=(2\1-2\2,-2\1+4\2,2\3) ...
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Definition of Global Convergence

I am confused by the notion of "global convergence" as used in numerical optimization literature, and did not find an exact definition for that yet. Now I try to double check my understanding here. ...
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Constrained Optimization of a function of two variables.

I was given the following tutorial problem, and I'm having a bit of trouble seeing how it works. I've been asked to find the four critical points of this system, with two of these being degenerate ...
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295 views

The minimum number of circles in order to obtain a COVER of a specific square

Suppose a unit square $X$, with side length $l=1$ as below, which is COVERed by a set $Y$ of circles with the same constant radius of $r=\dfrac{\sqrt{2}}{10}$, where a ...
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Conditional extreme value of a function

Let $x,y,z$ be the positive real numbers, if $x^2+y^2+z^2=1$, then how can we find the minimal value of this function $f(x,y,z)=\dfrac{xz}{y}+\dfrac{yz}{x}+\dfrac{xy}{z}$.
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Proof of a matrix is positive semi-definite

For $\ i = 0, 1, \cdots m$, $f_{i}(x): R^n \rightarrow R$ is defined to be $$ f_i(x) = x^TQ_ix + 2p_i^Tx + r_i $$ , where $Q_0 \cdots Q_m$ are real symmetric matrices, $p_0 \cdots p_m \in R^n$, and ...
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How to linearize a quadratic objective function with linear constraints?

I have an optimization problem that I'm working on. The objective is defined as follows: $Maximize: c_i\cdot w_i \cdot x_i - d_i \cdot y_i \cdot \delta_i $ subject to some linear constraints where ...
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127 views

Least squares with a quadratic inequality constraint

Is there a closed form solution for the following least squares problem: $$ \min_\mathbf{x}\|\mathbf{a+Bx}\|^2 ~~\text{s.t}~~\|\mathbf{x}\|^2 \leq \alpha^2$$ where $\mathbf{a} \in \mathbb{C^{M\times ...
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129 views

How to interpret a discontinuity in 2D Pareto Frontier?

I've solved a bi-objective optimization problem by means of NOMAD solver from OPTI Toolbox and as a result I've obtained a Pareto frontier: How to interpret the visible "gap" in the Pareto frontier? ...
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251 views

Unsolvable(?) Assignment Problem

I've recently been trying to implement the Hungarian Method in C++, and I've been using 5x5 matrices to test my program. Last night I came across a matrix which neither I nor my program can solve. Is ...
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How to introduce flat cost of flow over a node using mixed integer programming.

In the set up for the program we have a graph where we are trying to minimize the cost of sending flow over the arcs. I have formulated the following linear program. \begin{array}{ll} \text{minimize} ...
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Linear Algebra Question concerning the trace of a symmetric positive definite matrix.

The objective is to minimize the diagonal elements of a symmetric positive definite matrix. The expression of this matrix is a little bit nasty and its inverse is much easier to deal with. Can I claim ...