Questions tagged [nonlinear-optimization]

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 trying to solve particular problems.

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. Usually, non-linear optimization problems are much harder to solve than linear ones.

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The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is $g(f)$ strictly convex. My attempt, since $f$ is…
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What is the definition of a first order method?

The term "first order method" is often used to categorize a numerical optimization method for say constrained minimization, and similarly for a "second order method". I wonder what is the exact definition of a first (or second) order method. Does…
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Let $P(x)\in \mathbb{R}[x]$ be of degree $n$ and for any $x \in \left(0,1\right]$, we have $x\cdot P^2(x) \le 1$. Calculate $\max P(0)$.

Let $P(x)$ be a polynomial with real coefficient and with degree $n$ such that for any $x \in \left(0,1\right]$, we have $$x\cdot P^2(x) \le 1$$ Find the maximum of $P(0)$. Note: $P^2(x) = (P(x))^2$. Any idea how to start? I suppose we can write…
nonuser
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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 MathOverflow on 13.12.2014; the question has been solved…
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Fenchel dual vs Lagrange dual

Consider the Fenchel dual and the Lagrangian dual. Are these duals equivalent? In other words, is using one of the these duals (say for solving an optimization), would give the same answer as using the other one? I think the answer is no, but I am…
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Operations research book to start with

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 optimization, convex optimization and quadratic…
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Find $x,y,z>0$ such that $x+y+z=1$ and $x^2+y^2+z^2$ is minimal

How can I find $3$ positive numbers that have a sum of $1$ and the sum of their squares is minimum? So far I have: $$x+y+z=1 \qquad \implies \qquad z=1-(x+y)$$ So, $$f(x,y)=xyz=xy(1-x-y)$$ But I'm stuck from here. Hints?
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How do I solve $\min_x \max(c_1^Tx, c_2^Tx, \dots, c_k^Tx)$ for $\lVert x \rVert_2 = 1$.

Let $f(x) = \max(c_1^Tx, c_2^Tx, \dots, c_k^Tx)$. where $x, c_1, c_2, \dots, c_k \in \mathbb R^n$. What fast iterative methods are available for finding the (approximate) min of $f$ with the constraint $\lVert x \rVert_2 = 1$? Notes: $f$ is convex…
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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 the players bid some amount from the available amount…
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Scaling factor and weights in Unscented Transform (UKF)

I'm trying to implement the UKF for parameter estimation as described by Eric A. Wan and Rudolph van der Merwe in Chapter 7 of the Kalman Filtering and Neural Networks book: Free PDF I am confused by the setting of $\lambda$ (used in the selection…
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is nonlinear least square a non convex optimization?

linear least-squares are convex optimization. Are nonlinear least squares also convex optimization? Can someone please give some simple examples?
Mia
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When $\min \max = \max \min$?

Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be compact sets. Consider a continuous function $f : X \times Y \rightarrow \mathbb{R}$. Say under which condition we have $$ \min_{x \in X} \max_{y \in Y} f(x,y) = \max_{y \in Y} \min_{x…
user693
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what to do when the multivariable second derivative test is inconclusive?

What do we do when the second derivative test fails? How do we approach it, and is there a general method to further find whether a critical point is a maximum, minimum or a saddle point? For example, I'm asked to find all the critical points of the…
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Special version of Tonelli’s theorem

I am trying to prove this theorem. I have not find anything similar to it in the internet Special version of Tonelli’s theorem Assume that the function $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{R},\,\, g(x, \xi): [a,b] \times \mathbb{R} \to…
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Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by $f(x) = sup_{i \in I}f_i(x)$ for $x \in \Omega$…
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