# 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.

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### The tangent space $T_pM$ in terms of the gradient

Let $f: {\mathbb R}^n \to \mathbb R$ be a ${\mathbb C}^1$ function. The graph of $f$ is the surface $M :=\{(x, f(x)) \in {\mathbb R}^n \times \mathbb R | x \in {\mathbb R}^n\}$. Given an arbitrary ...
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### Solving second-order nonlinear ODE numerically

I am trying to solve for the function $p(x)$ which obeys the following: \begin{align*} p\left(x\right) = \left(e^{x} - \frac{1}{2} z \left(p\left(x\right)\right)^2\right)^\gamma \phi(x), \tag{1} \...
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I have the following problem $$\max_x ~~ \sum_k | a^H_k B x |^2 \\ \text{s. t.} ~~ x^H B^H B x \leq c \\$$ where $x\in \mathbb{C}^N$, $B \in \mathbb{C}^{M \times N}$, $M > N$. I know the ...
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### Integer programming : linearize product of constants given conditions

I have some constant values $c_i$ in $(0.5, 2)$. I also have binary variables $x_i$. For my integer program, for a particular constraint, I need to multiply only those $c_i$ when $x_i$ takes the value ...
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### Minimizing costs of a specific geometry shape

I have geometry mathematical problem. I have a shape that is made of a cylinder and two half spheres as to top and bottom. How can I minimize the cost of this shape when the volume is known and the ...
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### Solving nonlinear least-squares with first order Taylor expansion

I'm referencing this Wikipedia article. I understand that a Taylor expansion of a function $f(x)$ around $x = a$ can be given by $f(x) = \sum_{n = 0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$ ...
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### Equivalence of solutions in min and max optimization problems

Let us consider the following optimization problem: for fixed $y$ ($y$ can be vector, possible from compact set) find $x$ such that \begin{equation*} \begin{aligned} & \underset{x}{\text{...
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### Computing tangent cone from hessian?

In optimization, one often assumes LICQ or other constraint qualification to determine the tangent cone of a set by takeing the gradient of the constraint functions. If the constraint has a vanishing ...
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### Linear Objective Minimization on Intersection of Two Ellipsoid Surfaces

Let $D=\text{diag}(d_1,d_2,\dots,d_n)$ be a positive definite $n\times n$ matrix, $0\ne c\in \mathbb R^n,$ and $\alpha$ be a positive real number such that $\alpha \ne d_i$ for $i=1,2,\dots,n$. ...
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### how to process $\max(x,0)$ in optimization problems

my objective contains the forms of $\max(x,0)$ which can be written as { \begin{align} \mathop{\max}\limits_{{\bf{P}},\theta}\quad&R^{sec}_{tot}\\ \textrm{s.t.} \quad\: &C_{c,e} \le R_{...
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### If $x$ is real find the maximum possible value of $10^x-100^x$

According to the person who gave this question it apparently has something to do with the range of a quadratic expression. But I can't see the connection with a quadratic equation. So I tried to ...
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Let $1\leq m<M$ be and $\alpha_1,\dots,\alpha_n>0$ be fixed real numbers. I want to solve the following $n$-dimensional optimization program \begin{aligned} \operatorname{min}&\, \sum_{i=... 1answer 69 views ### How can I linearize this inequality? [closed] Is there a way to linearize this inequality? maybe by separing this inequality into three? I have an optimization model where one of the restrictions is the following, which makes the model a non-... 0answers 13 views ### Linearizing nonlinear constraints with square term I am trying to solve an optimization problem with the following constraints. \begin{align} & n_m = \sum_{i=1}^I x_{im}T_i \quad \forall m\\ & n_m^{'} = \sum_{i=1}^I x_{im}C_i \quad \forall \\... 0answers 36 views ### Nonlinear least squares uniqueness Suppose I have a nonlinear least squares objective function I want to minimize: \chi^2(\mathbf{x}) = \sum_{i=1}^n f_i(\mathbf{x})^2 = \mathbf{f}(\mathbf{x})^T \mathbf{f}(\mathbf{x}) Now suppose ... 1answer 40 views ### What did Nemirovski and Yudin actually do in their 1978 article problem complexity and method efficiency in optimization? What did Nemirovski and Yudin actually do in their 1978 book problem complexity and method efficiency in optimization? I'm struggling to find very much on it. 0answers 18 views ### Optimization Method Solver I need to maximize r(x)=0.6452(1-x)+2.98*10^{-4}\ \frac{(1-x)^{3}}{x^2} wrt to x, which lies between 0&1.\ 1.) Which optimization technique to use? 2.) What is the optimal value of x? 3.) Is the ... 0answers 29 views ### Corner Solution To A Recursive, Strictly Concave Function? I was reading through Dynamic Programming by Richard Bellman today, and I got to exercise 7 in chapter one. You are asked to prove a theorem, but I feel like the theorem itself is... well, not quite ... 0answers 47 views ### Quadratic programming minimization problem I'm trying to minimize a function but it may be beyond my ability. Can you help me go through it? Do you have any advice on how this can be solved? Let X be a finite set \{x_1, ..., x_n\} of ... 0answers 24 views ### “Batchwise” least squares with smoothness in row direction as extra objective My math background is essentially non-existant, so please bear with me. I have a "batchwise" (for lack of a better term) linear least squares problem A X = Y that I solve like \hat X = A^\dagger Y... 0answers 21 views ### Linearize a constraint that has division of sum of binary variables I am trying to solve an optimization problem with the following constraints. \begin{align} & n_m = \sum_{i=1}^I x_{im}T_i \quad \forall m\\ & n_m^{'} = \sum_{i=1}^I x_{im}C_i \quad \forall \\... 1answer 19 views ### Is the difference of a convex function and a strictly convex function convex? Given two functions, f which is convex and g which is strictly convex, is the difference f-g convex? My impulse is to say no, since -g should be concave, but I'm trying to show this ... 0answers 16 views ### max decomposition I am looking to solve a problem of this form: \operatorname{argmax }_x\{f_1(x)-\operatorname{argmax}_yf_2(y)\} I am wondering if there any related math that we can decompose this onto something ... 0answers 20 views ### Augmented Lagrangian method for standard form Linear Program What will be Augmented Lagrangian equation and its derivative for standard form LP. minimize c^Tx subject to Ax = b x ≥ 0 I have tried but not sure wether it is correct or not. L_\rho(x,\lambda,... 0answers 15 views ### Contraction property of scaled gradient descent Let f(x):\mathbb{R}^n \to \mathbb{R} be \sigma-strongly convex and L-Lipschitz continuous. Assume to apply a scaled gradient descent to find the minimum x_\star of f, i.e., x_{k+1} = x_{...
I have the following optimization problem in $x \in \mathbb{R}^n$ \begin{array}{ll} \text{maximize} & u^T x - c \sqrt{x^TAx}\\ \text{subject to} & \sum_{i} x_i = 1\\ & x_i \geq 0\end{...
Consider the optimization problem f(x) = infimum $(−x^2) s.t. 0 ≤ x ≤ 1$. What will be its dual in simplified closed form with no 'inf'. I know its Lagrangian function will be \$ L(x,𝜆) = -x^2 +𝜆...