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.

learn more… | top users | synonyms (5)

1
vote
0answers
45 views

how to minimize this convex function?

$x_i$ and $y_j$ are variables. I intend to minimize this function and obtain the optimal value of $x$ and $y$: $\begin{align} ...
1
vote
2answers
169 views

Applying the Lagrangian function to find critical points

So I have the following function $$ f(x,y) = x^2+y^2 $$ subject to $$ g(x,y) = x+y-1 = 0. $$ And I have to use the Lagrangian to find the critical points, and determine wether they are ...
1
vote
1answer
150 views

What's a really good book for a course titled “Optimization and Control Theory”?

I can't seem to find one that shows a lot of examples with the theory. Could I get some help? Also, it would be a bonus if the book/material is readily available online so I can download it onto my ...
1
vote
1answer
337 views

Determine all the extrema of a function subject to a non-linear constraint.

QUESTION Determine all extrema of the function $$f(x,y) = x+ 2y $$ subject to $$x^2 + y^2 - 80 = 0$$ ATTEMPT I don't think I understand what I'm supposed to do. This was in a test and I ended up ...
1
vote
1answer
793 views

How many n square can fit into a square of side N

Suppose we have n small squares of equal sizes that has area w. Suppose we have a fix square S of area A such that for area A, one area w < area A. If square S's area A, length, and width are ...
2
votes
1answer
296 views

Minimize $\|Ax-b\|$ where $x$ is a binary vector

For a software project I'm involved on, I have a situation where I have a large vector that is the sum of some smaller vectors. I know all the possible small vectors, and I know that no two of them ...
0
votes
2answers
199 views

This question is a basic optimization problem, also a linear algebra question:

Let $p$ be a direction of unboundedness for the constraints $$Ax = b, x ≥ 0.$$ Prove that $−p$ cannot be a direction of unboundedness for these constraints.
0
votes
1answer
21 views

Geometric interpretation of a critical point, i.e. of $q(t) := f(x + t(y-x))$.

So, I know what critical points are. But hear me out on the following notes I made: For $x,y\in \mathbb{R}^n$ we define $$q(t) := f(x + t(y-x)), $$ then $$q'(t)=\nabla f(x+t(y-x))^T(y-x).$$ Now, if ...
-1
votes
1answer
44 views

Can someone help me please?

$A^+A=I$ where $A^{+}=(A^TA)^{-1}A^T$, $A_{m \times n}$ I have tried with $(A^TA)^{-1}=A^{-1}{A^T}^{-1}$ but the matrix is not squared
1
vote
1answer
34 views

For any positive integer $n$, what is the value of $t^*$ that maximises the following expression?

For any positive integer n, what is the value of t* that maximises the following expression? $$\displaystyle \sum_{j=1}^{n-t^*}\left(\frac{t^*-j+2}{t^*+j}\right)$$ where $t^*$ is some integer in ...
0
votes
1answer
29 views

On a quadratic optimization

The problem is formulated as follows: Given $0\neq x \in \mathbb{R}^n$, and $k\leq n$, consider the following optimization problem $$\min_{\textrm{rank}(C)=k}x^t(I_n-C)^t(I_n-C)x$$ where $I_n$ be the ...
-1
votes
1answer
278 views

Sum-to-one constraint

This is a general question, but I am asking it since I am not able to find any good material online. Can someone please explain what's meant by a "sum-to-one constraint"? Thanks.
1
vote
2answers
522 views

Distance between point and sine wave

I have a project where I need to know the exact minimal distance between a point $(e, f)$ and a sine wave $y = a + b\cdot\sin(cx+d)$ Is there any way of calculating this? If not, is there a way to ...
1
vote
1answer
56 views

Add and subtract in optimization

I have a problem related to the use of the "add and subtract" strategy in optimization problems. This is related to a question I asked on Cross Validated. I got really helpful answer to this question, ...
1
vote
1answer
388 views

Equivalence of Maximizing Products, Sums, and Sum of Logs

Throughout this question assume $f_i \ge 0 \forall i $. I know that for any (single) function the following is true $$f(x^\star) \ge f(x) \text{ }\forall x\in X$$ iff $$\log(f(x^\star)) \ge ...
0
votes
1answer
49 views

Second-order Lagrange condition

I am not sure what the second-order Lagrange condition is and how it applies to this? Minimize $x^2 + y^2$ Subject to $x^2 - y - 4 \leq 0$ and $y - x - 2 \leq 0$. Please can someone assist me in ...
3
votes
1answer
56 views

How to find the maximum of $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a+b}+\dfrac{1}{a+c}$ given certain constraints.

Let $a,b,c\ge 0,$ and such $a+b+c=1$. Find the maximum of: $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a+b}+\dfrac{1}{a+c}$$ My try: ...
1
vote
1answer
43 views

Proving certificate of inequality

I have a question about proving the certificate of inequality in the given question: If there exists $y$ such that $y^T A \leq 0$ and $y^T b < 0$, then $Ax = b$, $x \leq$ 0 has no solution. I ...
0
votes
1answer
55 views

Inequality Constrained Optimization Problem

I am working on the question displayed below. I am not sure if I understand it correctly and I am looking for some input. So, I am asked Why is $x^*$ a local maximum for $f$ subject to the set ...
0
votes
1answer
141 views

Reference request - second derivative test for function of two variables that includes details of what you can infer when discriminant is zero

The second derivative test for functions of two variables as I have learned and taught in calculus classes says, in part, that if at a point $D=f_{xx}f_{yy}-(f_{xy})^2$ is zero then we can tell ...
3
votes
1answer
238 views

Attain minimum on boundary of a convex set?

It is well known that there exists a unique minimum norm vector over a closed convex set. Suppose we have a Banach space X (if it needs to be more concrete we can think of $L_2$, the space of square ...
1
vote
1answer
42 views

Minimization of $\text{tr} (W^TMW)-\text{tr}(NW)$ subject to $W^TW=I$

Is there a closed-form solution for finding W that minimizes the objective function: $\text{tr} (W^TMW)-\text{tr}(NW)$ subject to $W^TW=I$ where $M$ and $N$ are fixed matrices. I find it difficult to ...
2
votes
1answer
167 views

Projectile Trajectory with Air Drag

Given the following equations and values, Find an initial theta value to maximize horizontal range with air drag. $f(x)=\tan(\theta)*x-16+(x/(200*\cos(\theta))^2$: height with no air drag ...
0
votes
0answers
140 views

Multi objective optimization into single objective.

I read that it is possible to convert a multi-objective optimization problem into single objective by using weighted sum method. I wanted to know if it is a good idea to convert a two objective ...
1
vote
2answers
1k views

Zero Eigenvalues for Hessian Matrix

I need to show that along any line passing through the origin, $$F(x,y) = 3x^4 -4x^2y + y^2$$ has a minimum at $(0,0)$ but that without the restriction, there is no local minimum at $(0,0)$. The ...
4
votes
2answers
220 views

Maximizing a ratio of convex matrix functions by minimizing a difference?

Given that $g(.),h(.)$ are twice-differentiable convex quadratic real functions whose domain is the set of all real matrices while the range is the set of positive real numbers, then: Is maximizing ...
0
votes
2answers
440 views

For the function $f(x) = x^2 \sin(1/x)$ for $x \ne 0\,$and $f(x)= 0$ if $x=0$, show that $f$ does not have local maximum or minimum in $0$.

We have $f(x) = x^2 \sin(\frac{1}{x})$ for $x$ does not equal $0$ and $f(x)= 0$ if x equals $0$. I know that the function is continuous for all $x$ by using the squeeze theorem. I also know that in ...
0
votes
1answer
47 views

Optimization with inequality constraints

Could someone give a solution to this problem as well as an interpretation of the results? I am not sure how to deal with inequality constraints. Part 1: Find the minimum value of $f(x) = |x|^2$ ...
0
votes
3answers
49 views

how to solve this equation containing “min”?

$$ xy=128,x+y=\min $$ How to find $x$ and $y$ with the minimum sum? This example is simple and can be done by brute forcing but I want to know what is the proper way of solving it. How to solve ...
0
votes
1answer
58 views

Find minimal $\alpha_3$ such that $u\in H^3(\Omega)$ and $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$

My instructor presented me the quiz below but forgot to define key terms such as minimality and $H^3$. Quiz Let $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$. Find the minimal $\alpha_3$ such that $u\in ...
0
votes
1answer
32 views

Minimum of function

Let $f:\mathbb{R}\to\mathbb{R}$ be a function with $f(x)=\dfrac{(x^4-2ax^3+3a^2x^2-2a^3x+a^4+9)}{(x^2-ax+a^2)}.$ Determine the minimum of the function, if we know, that $-2\leq a\leq2$, $a\neq0$. I ...
1
vote
0answers
213 views

Strange but practical Bin packing problem

I am trying to solve the following MILP through LP solve. A link for the original problem is here I am re-iterating the problem as follows: I am trying to write an application that generates drawing ...
9
votes
2answers
275 views

The fastest trajectory that a particle should follow through two different mediums

Hello, after 3 failed attempts to solve this problem, I decided to start a bounty for this question. Please, I need a complete answer with an interpreation of the final result. Thank you in advance. ...
0
votes
1answer
247 views

Linear Programming - Overtime restriction

hopefully I can get some help on this problem, it's got me quite stumped. I was given a linear programming problem with the goal of minimizing labor costs. The variables x_t represent the number of ...
5
votes
1answer
1k views

Find lower bound of function

Can someone help me finding a lower bound to the function $$f(x)=\frac{x-1}{e^{-1}-xe^{-x^2}},$$ where $x\in[1,+\infty[$? Taking the derivative and then solve $f'(x)=0$ isn't analytically possible. ...
0
votes
1answer
27 views

What is the value of $[c,d]$ when $c$ and $d$ be such that $f(x) ∈ [c, d]$ for all $x ∈ [a, b]$?

Let $c$ and $d$ be such that $f(x) \in [c, d]$ for all $x \in [a, b]$. What is the value of $[c,d]$ for the function $f(x)=\sqrt{1-x^2}$ on the interval $[a, b]=[0,1]$? I knew taking the minimum and ...
1
vote
1answer
349 views

Business Linear Programming Question

Now I don't need you guys to do my homework for me; however, I am a little stumped Xara Stores in Canada imports the designer-inspired clothes it sells from suppliers in China and Brazil. Xara ...
0
votes
1answer
333 views

Algorithm for estimating $\beta$ using a Taylor series expansion

I am working on the following question for a mathematical economics class. Consider an econometric model: $$y_t=f(x_t,\beta) + e_t,t=1,...,T$$ where $\{ e_t \}$ is a sequence of mean-zero ...
0
votes
0answers
42 views

On the solution of a stacking/spending optimisation problem

Description of the problem Each week our hero, say John, can stack a minimum of $2.5$ hours and a maximum of $6.5$ hours if he works more than $8$ hours per day. Each day John cannot stack less than ...
1
vote
0answers
58 views

When using lagrange multipliers why does computing $\nabla f = \lambda \nabla g$ give a mix or a min ( I do not understand the concept entirely)

Why should it be that when $\nabla f$ and $\nabla g$ are parallel, that at this point there is a max or a min?
0
votes
1answer
123 views

Mathematical formulation for maximum sum of edge weights

For my academic research purposes, I have a situation as below. The initial problem looks as in below figure. I need to find one match for each of P1,P2,P3 from the right side such that the sum of ...
1
vote
1answer
150 views

Simultaneous Maximization and Minimization

I have a function with two variables say $$g(x,y)=f(x)−h(x,y)\ $$ where $$ f(x)= ax-bx^2\ $$ and $$h(x,y)=(x+y)^2\ $$ and $$ y>=0, x+y>=0\ $$My purpose is to maximize g(x,y) for x, ...
0
votes
1answer
47 views

Conditions for a system to be solvable.

I have the following system of equations: $$\begin{aligned} \left\{\begin{array}{l} a+dz+cy+exy = 0\\ 10a+3bx-exy =0\\ -5a-dz = 0 \end{array}\right. \end{aligned}~~.$$ I would like to solve for ...
1
vote
1answer
57 views

Maxima problem related to physics

I was solving a physics question, and in the end, I got it down to 'finding the relation between H and h such that the following expression is maximized': $\sqrt{H-h} + \sqrt{h}$. Can you please ...
0
votes
0answers
100 views

A basic question on contours/level sets of a function in quadratic form

Consider $$f(x)=\frac{1}{2}x^TQx-x^Tb$$ where $Q$ is an $n \times n$ symmetric matrix The contours of $f$ are $n-$dimensional ellipsoids with axes in the directions of the $n$-mutually orthogonal ...
3
votes
2answers
137 views

Maximize the determinant

Over the class $S$ of symmetric $n$ by $n$ matrices such that the diagonal entries are +1 and off diagonals are between $-1$ and $+1$ (inclusive/exclusive), is $$\max_{A \in S} \det A = \det(I_n)$$ ...
1
vote
0answers
51 views

Regularity questions in constrained variational problem

Consider the problem of minimizing $$ I(u) = \int_a^b F(t,u(t),u'(t)) d t $$ over, say, $W^{1,\infty}(]a,b[)$. Then regularity theory tells us that if $F$ and $F_{\dot q}$ are $C^k$, and in addition ...
1
vote
0answers
29 views

Maximum value for $F(y)=\int_0^1[y'\sin(\pi y)-(y-t)^2]dt$?

This is what I have done thus far: $F(y)=\int_0^1[y'\sin(\pi y)-(y-t)^2]dt=-\frac{1}{\pi}\int_0^1[(\cos(\pi y))\frac{d}{dt}]dt-\int_0^1(y-t)^2dt$ (as $-\frac{1}{\pi}[(\cos(\pi ...
2
votes
1answer
93 views

Interchanging the order of maximization and diffrentation

Let $f(x,y):\mathbb{R\times\mathbb{R}\mapsto\mathbb{R}}$ be a continuous and differentiable function. When can we claim that the following holds true: $$ \frac{d}{dx}\max_{y\in\mathbb{R}}f(x,y) = ...
2
votes
0answers
96 views

The most efficient algorithm to solve the following problem

Is there an efficient optimization algorithm to solve the following problem? $(\alpha,\beta,\gamma,\cdots) =$ argmax $\sum_{i}\log(\alpha a_i+\beta b_i+\gamma c_i+\cdots)$, s.t. ...