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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McCormick Envelopes with more then 2 variables

I'm trying to solve a bilinear optimization problem by linearizing the problem using the McCormick Envelope method. It's quite a simple method when you are only using the product of two variables, ...
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Distance between a point and a conic curve

I have a point $r=(100,0)$ and want to find the closest point to it from this set: $$k = \{(a,b) : b^2=1+a/4\}$$ where $a$ belongs to $[-4,0]$. I thought about defining function $h(x)=|r-x|$, and ...
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799 views

Optimization: maximum area of a triangle under a parabola

Optimization: maximum area of a triangle in a parabola Inside a curve ($x^2-25$ - Parabola) a triangle is drawn with A as the vertex at the origin and the line joining points B and C lie on the ...
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Logistic regression for football results - Estimating coefficient through maximum likelihood

Consider two football teams $V$ and $L$ with strengths $W_V$ and $W_L$, respectively. Let's assume that the draw probability $\mathbb{P}(Draw)$ is known. Then this model is supposed to give estimates ...
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3answers
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Can't find minimum using Lagrange multipliers

I want to find the minimum of the function $f(x,y) = x + y^2$ with the constraint $2x^2 +y^2 = 1$. Here are my partial derivatives: $$f_x = 1$$ $$f_y = 2y$$ $$g_x = 4x$$ $$g_y = 2y$$ I have the ...
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How write one optimization formula.

In this game I start with a Galleon with capacity 400. I can upgrade the harbor to get more Galleons, or upgrade the technolgy to increase the Galleon base cargo size by 10%. Right now have 8 ...
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35 views

What are spurious local optima?

I keep seeing that word "spurious" (when used in the context of optimization), but I'm having trouble finding a good reference on what the definition of the term is.
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18 views

floor/ceiling/round functions in the constraints of an optimization?

I have a constrained optimization problem in which I have to impose a "floor" or "ceiling" constraint to the solution. In fact I decided to use these nonlinear rounding functions because I needed to ...
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28 views

Lasso with non-linear objective

I have a non-linear objective function that I want to minimize considering some constraints in order to obtain a sparse solution (lasso type). min f($\theta$) s.t. $\sum_i|\theta_i|\leq t$ $\theta_i ...
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1answer
45 views

Why does the “printing neatly” algorithm use cubes rather than squares?

In Introduction to Algorithms, 2nd ed. (Cormen, Leiserson, Rivest, and Stein), ch. 15, Dynamic Programming, problem 15-2 Printing neatly (a copy of which is here), the official solution given in ...
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15 views

Maximum (edge)weight connected subgraph of an undirected graph.

Let G be a undirected graph with weighted edges. I want to find a connected subgraph which has at most L nodes(vertices) whose sum of edges is maximum. It sounds similar to MWCS or PCST but here only ...
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Convergence of the minima of functionals

Let $\mathcal{H} \subset \mathbb{R}^3$ denote a compact subspace. Suppose we have a sequence of functionals $(Q_n)_{n\geq 1}$ and a functional $Q$ from $C(\mathcal{H},\mathbb{R}^3)$ (which is the ...
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Multi-objective optimization or single objective optimization?

I have this function: A(x)= P(x) / B(x) Firstly I thought about doing an multi-objective optimization, maximizing A(x) and minimizing B(x) because this two values are very important. But if I just ...
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12 views

Does projected gradint descent(pgd) results in the same minimizer as the one given by unconstrained gd and projected back on the constrained set?

For $f: \mathbb{R}^n \mapsto \mathbb{R}$ with $f(x) < \infty,\;\forall x \in \mathbb{R}^n$ and for convenience let's assume $f$ is continuously differentiable. Suppose we are trying to solve the ...
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Minimizing a strictly convex function with inequality constraint

So we've been learning about the Kuhn Tucker conditions in my non-linear optimization course and I've been having trouble with this problem: QUestion: description here Question: a strictly convex ...
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4answers
82 views

Minimize the function $f(y_1,y_2)=3 y_1^2+8y_2^2$ [closed]

I would like to minimize $f(y_1,y_2)=3 y_1^2+8y_2^2$ with the constraints $g(y_1,y_2)=y_1^2+y_2^2=1$. I thought I could use the Lagrange multipliers, but it is not work. Is there anyone could show me ...
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96 views

Least squares problem with constraint on the unit sphere

It is easy to find the minimum of $\|Ax-b\|_2$, when $A$ has full column rank. But how is the case when we add the constraint $\|x\|_2=1$? Or, to be explicit, $$\min_{\|x\|_2=1}\|Ax-b\|_2=?$$ My ...
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Optimal number of operations in the given scenario?

Suppose $$A_1=\{x_1+x_2+x_3,\quad x_2+x_3+x_4,\quad x_3+x_4+x_5\} \\ A_2=\{x_0+x_1+x_2, \quad x_0+x_1+x_8, \quad x_0+x_7+x_8\} \\ A_3=\{x_{10}+x_{11}+x_{12}, \quad x_{11}+x_{12}+x_7, \quad x_7+x_8+x_{...
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deriving Newton's method for optimization

I thought I understood the derivation of Newton's method for finding a minimum, but just realized I was not being at all careful! Here are three alternate "derivations". I think the first two are ...
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Basic Question about Newton's Method for Optimization

This is a very basic question about Newton's method for optimization, but I cannot seem to find the answer in any of my searches. If we are using Newton's method (or gradient descent), how do we find ...
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1answer
673 views

How to solve a quadratically constrained linear program (QCLP)?

Can anybody suggest some techniques to solve a quadratically constrained linear program (QCLP)? Any references on standard techniques would be helpful.
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Change of variables in minimization

I have the following non linear programming to solve: $$\left\{\begin{matrix} \min & (x-y)^2 +e^z+e^{-z} \\ \text{s.t.} & xz=0 \\ & yz=0 \end{matrix}\right.$$ The book suggests to ...
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Minimizing $\cot^2 A +\cot^2 B + \cot^2 C$ for $A+B+C=\pi$

If $A + B + C = \pi$, then find the minimum value of $\cot^2 A +\cot^2 B + \cot^2 C$. I don't know how to solve it. And can you please mention the used formulas first. What I can see is that if one ...
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323 views

What happens if we remove the non-negativity constraints in a linear programming problem?

As we know, a standard way to represent linear programs is $$\begin{array}{ll} \text{maximize} & c^T x\\ \text{subject to} & A x \leq b\\ & x \geq 0\end{array}$$ with the associated dual ...
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A question about Lagrange multiplier in optimization

I read @amoeba 's answer in this post, PCA optimization problem is $$ \underset{\mathbf w}{\text{maximize}}~~ \mathbf w^\top \mathbf{Cw} \\ \text{s.t.}~~~~~~ \|\mathbf w\|_2=1 $$ where $\mathbf C$ ...
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KKT conditions for different inputs.

So I have the following problem: I'm trying to get a demand function for a nonlinear 2 variable optimisation problem. There are 3 inequality constraints. Doing the usual thing I get the following ...
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Book Recommendation for Infinite Dimensional Stochastic Optimization Problem in Discrete Time

Let $X(k)$ be i.i.d. discrete random variables and for all $k=0,1,2,...,N-1$, let $X:=(X(0),X(1)...,X(k))$ and $f := (f(0), f(1),...,f(k))$ with $f(k)$ be the decision function at time $k$, I want to ...
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Why the original MINLP and Linearized MILP are giving mismatched results?

I have an MINLP and its linearized formulation problem given below where the objective (nonconvex) and constraint C4 are nonlinear. We linearized them by applying some known techniques. However, when ...
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Distance from a set to a point

There is this exercise I cannot understand well. It asks me for the distance between this set in $\mathbb{R}^3$ $$U = \{(x, y, z)\ |\ ax + y - 2z = 0, z = 0 \}$$ and the point $(0, b, 1)$. Also it ...
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Minimising a Loss Function requiring Matrix by Matrix Derivative

I am trying to minimise the following cost function with respect to $X$: $\mathbf{C}(X) = ||{M \cdot X \cdot \mathbf{1}_{N \times 1} - T}||_{2}^{2}$ Here, $M$, $X$, $T$ are matrices of dimensions $a \...
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Extremas on multivariable functions

if the gradient Of a function f(x,y,z) has all its partial derivatives (fx, fy) at a point p equal to zero but the partial derivative z at that point is equal to a constant i.e fz= 12. In this case ...
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Given a population of fish with exponential growth, what is the optimal strategy for fishing?

Suppose we have a population of fish, say $10000$, with an exponential growth each year of $30\%$. If we want to collect as many fish as possible in, say 10 years, a natural question to ask is: ...
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Minimizing trace with equality constraints

I would like to solve the following trace-minimization under equality constraints optimization problem: $$W^* =\arg\min \operatorname{Tr}[WCW^T] \text{ s.t. } A=B^TW^TWB$$ where $W,C\in\mathbb{R}^{...
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How to prove Lagrange multiplier theorem in a rigorous but intuitive way?

Following some text books, the Lagrange multiplier theorem can be described as follows. Let $U \subset \mathbb{R}^n$ be an open set and let $f:U\rightarrow \mathbb{R}, g:U\rightarrow \mathbb{R}$ be $...
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Inverse Vectorization Vec^-1

Hope that you will find this post in good health. I am Mr. Adnan from Pakistan with research background in Control systems. I am working on one problem in which Hadamard weights are using. During ...
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Maximum value of $\sin A+\sin B+\sin C$?

What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin ...
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Prove that $\sin A+\sin B+\sin C\leq \frac{3}{2}\sqrt3$ [duplicate]

If $A,B,C$ are the angles of a triangle, prove that $\sin A+\sin B+\sin C\leq \frac{3}{2}\sqrt3$ I want to prove this inequality without Jensen's inequality, as Jensen is not in my syllabus. Let $\...
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Maximizing area of a pentagon

Suppose $a,b,c,d,e$ are pairwise distinct positive integers. Consider a pentagon with sides $a,b,c,d,e$ and with angles maximizing its area (we assume that a pentagon with such sides exists). It is ...
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Failure of second derivative test of two variable function where the all the partial derivatives are equal

Actually I am looking to find the local minimum of the following function : $$F(x,y)=\frac{\Gamma(x+y+1)\Gamma(n-x-y+1)}{\Gamma(n+1)}$$ The partial derivatives of this function are: $\begin{align} ...
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Upper bound for $\gcd(a,b)$ if $\frac{a+1}{b}+\frac{b+1}{a}\in\Bbb{N}$

Suppose that $a,b$ are two positive integers so that $\frac{a+1}{b}+\frac{b+1}{a}$ is also a positive integer.Find the best upper bound for $\gcd(a,b)$. My work: $\frac{a+1}{b}+\frac{b+1}{a}=\frac{...
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Individually checking constraints for convexity in Optimisation problem valid?

I have a quadratic minimisation problem where both the objective fn and constraints have some quadratic terms. (Such as a throttle variable (continous) * On/Off (integer variable)). My question is: ...
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Minimization of vector sum through rotation

I'm curious if there's an algorithmic way to find the minimal vector sum of $N$ vector magnitudes by applying a rotation $\Phi$ to each individual vector. As an example in two dimensions, if I have ...
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Minimize $-\sum\limits_{i=1}^n \ln(\alpha_i +x_i)$

While solving PhD entrance exams I have faced the following problem: Minimize the function $f(x)=- \sum_{i=1}^n \ln(\alpha_i +x_i)$ for fixed $\alpha_i >0$ under the conditions: $\sum_{i=1}^n ...
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What are some applications of vertex separators?

What are applications of finding a vertex separator that minimizes a cut in a graph. To clarify the problem I am talking about is is given a graph of n vertices and a partition $m_1,m_2,..,m_k $of ...
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27 views

Robusness of median

If we let $X$ be a set of pints in $\mathrm{R}^2$, and let $g(X) = \arg \min_{y \in \mathrm{R}^2} \sum_{x_i \in X} \parallel x_i -y \parallel_2$ (geometric median of $X$). If $X$ and $X'$ are ...
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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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Maximum und Minimum [closed]

I need help for c) also I solved a and b, but c) I could not. Let $f~:~\Bbb R^2\to \Bbb R$ be defined as $f(x,y)=e^{-x^2-y^2+2x-2y}$ a) Determine all local extrema for $f$. b) Determine all ...
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Optimization of function on integer hypertetrahedron

I have the following optimization problem: Let $k,n \in \mathbb{N}$ with $k < n$. Let $N:=\{1, \dots,n\}$ and $D := \{^{t}(x_1, \dots, x_k) \in N^k \vert \sum_{i=1}^k x_i = N\}$ (hypertetrahedron)...
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Extrema on (compact) vinculum

My textbook ask to find the extrema of $f(x,y) = 2x^2+y^2$ on $x^4-x^2+y^2-5=0$. It uses the lagrangian multipliers to find critic points.. Then it computes the function on these points then says "...
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1answer
479 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...