# Tagged Questions

A non-linear optimization problem includes an objective function (to be minimized or maximized) and some number of equality and/or inequality constraints where the objective or some of the constraints are non-linear. Use this tag for questions related to the theory of solving such problems or for ...

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### Maximizing curious symmetric function from simple combinatorics

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). (This question has been posted at ...
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### Bidding Tic Tac Toe

In regular tic tac toe, both the players get alternate chances. This is a variant of that. Player $A$ has $\$x$amount and player$B$has$\$y$ amount as initial balance. Assume that $y>x$. Both ...
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### finite polynomials satisfy $|f(x)|\le 2^x$

This is a problem from TsingHua University math competition for high school students. Prove there exists only finite number of polynomials $f\in \mathbb{Z}[x]$ such that for any $x\in \mathbb{N}$ ,...
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### A numerical optimization problem with a convolution in the constraint

I have a problem of the following form: minimize $\|Dx\|_2$ subject to $\|x*x\|_2 = 1$ where $x\in\mathbb R^n$, $D$ is a given diagonal matrix of positive entries, and $*$ represents convolution, ...
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for somebody having a quite strong background in Mathematics, which are some good books for the domain of Operations research? I guess there are textbooks covering topics like linear and nonlinear ...
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### Optimization problem for a parity-check code

I have $n$ data blocks and $k$ parity blocks distributed across $m$ boxes where each box can contain atmost $b$ blocks. Each parity block is Ex-or of some data blocks (for ease of understanding we can ...
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### Minimization of $\sum \frac{1}{n_k}\ln n_k >1$ subject to $\sum \frac{1}{n_k}\simeq 1$

Looking at an algorithm for minimizing $\sum_{k=1}^{m} \frac{1}{n_k}\ln n_k > 1$ subject to $\sum_{k=1}^{m}\frac{1}{n_k} = 1$ in which $n_k$ are positive and in general non-sequential integers, I ...
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### Consider the problem minimize $f(x)= x^4 −1.$

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
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### Minimizing with Lagrange multipliers and Newton-Raphson

I am writing a program minimizing a real-valued non-linear function of around 90 real variables subject to around 30 non-linear constraints. I found handy explanation in CERN's Data Analysis BriefBook....
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### Are the Karush-Kuhn-Tucker conditions applicable when one or more of the constraints are nonlinear?

I am just beginning to read about the use of "Concave Programming" methods and use of the Karush-Kuhn-Tucker conditions to identify the maximum value of a non-linear objective function subject to ...
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### A naive approach for nonlinear optimization on several real variables and one natural variable. Give examples of (potential) failure

Motivation: Example. To solve a problem on evaluating the maximum of a product of $n$ real variables subject to an equality constraint on its sum $S$ ($=100$), I used the Lagrange multipliers method (...
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### Finding an explicit expression for a minimizer

Suppose $f$ is a continuous function on the interval (0,1). We consider the energy functional $F(u) = \int^1_0\frac{1}{2}((u')^2+u^2)\,dx - \int^1_0fu\,dx$ which is well defined for continuously ...
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### Kuhn-Tucker condition is not satisfied

Show that the solution to finding minimum of $f(x)=-x_{1}$ With conditions $-\sin(x_{1})+x_{2} \leq 0$ $x_{1}-x_{2} \leq 0$ is point $(0,0)$, but the Kuhn-Tucker condition is not satisfied in this ...
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### Nonlinear Optimization/Programming: A good counter text

I am currently taking a nonlinear optimization course and the text is Bertsekas' "Nonlinear Programming 2e". I think the book does a decent job but I am a much more "hands-on" and visual learner so ...
Find the smallest $\alpha\in\mathbb{R}$ such that, for all $x,y,z\in\mathbb{R}$, the following inequality holds $$\alpha\,\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)\ge(xyz)^2+|xyz|+1\... 1answer 2k views ### Intuition for gradient descent with Nesterov momentum A clear article on Nesterov’s Accelerated Gradient Descent (S. Bubeck, April 2013) says The intuition behind the algorithm is quite difficult to grasp, and unfortunately the analysis will not be ... 1answer 90 views ### P_{1c} = AP , P_{2c} = BP. How to find P? (being that A and B are 3\times 4 matrices and P is a 4\times 1 vector) This problem arose in my stereo vision project.$$ P_{1c} = A*P  P_{2c} = B*P $$where: P_{1c} and P_{2c} are 3\times1 vectors, A and B are 3 \times 4 matrices and P is a 4\times1... 1answer 203 views ### Monotonic Function Optimization on Convex Constraint Region So I have the following function, which I want to maximize:$$f(x_1,...,x_n) = \sum_{i=1}^n\alpha_i\sqrt{x_i}$$(where all \alpha_i are positive), subjected to the following equality and inequality ... 1answer 60 views ### Stuck on step in Lagrangian Problem For w,E column vectors, i the vector of ones, and \Sigma - an n\times n positive definite symmetric matrix, I am trying to solve the following maximization problem:$$ \max_{\{ w\}} \left\{ \...
Consider the parametric optimal solution $x^{*}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ defined as  x^*( y ) := \arg\min_{x \in X } \ \ x^\top x + x^\top A y \\ \quad \qquad \text{subject to: } \ f(...