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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Quadratic objective with quadratic constraint with projection on a matrix

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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25 views

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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Existence of solution of generalized variational inequality

I am studying the course Variational inequality and optimisation in analysis,we have the following definition: Definition:Let $K$ be a nonempty subset of $\mathbb{R^n}$, $F:K\rightarrow 2^\mathbb{...
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0-1 non convex quadratic problem in CPLEX [closed]

Can CPLEX solve non-convex 0-1 quadratic optimization problem? If so, does anyone know how?
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Solving $max \sum_{i=1}^2 \sum_{j=1}^2 (\phi_i A_{ij} \psi_j)^2$

Could you please provide some hints for me to solve this optimization problem? Here, for any $i=1,2$ and $j=1,2$, $\phi_i$ and $\psi_j$ are unknown vectors, $\alpha_i$, $\beta_j$ are some known ...
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Nonlinear optimization algorithms that consider evaluation cost

I'm familiar with a wide variety of local and global nonlinear optimization algorithms and the numerical libraries that implement them (such as NLopt https://nlopt.readthedocs.io/en/latest/). In my ...
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Explain why the second-order optimality conditions are unable to resolve this case

Consider the Problem $$\textrm{Minimize} \ \ \{(x_1 -1)^2 +x_2^2; \ \ g_1(x)=2kx_1 -x_2^2 \leq 0\},$$ for the case $k = 1$. 1) Provide an analytical argument to show that $X =(0,0)^{'}$ is an ...
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Nonlinear optimization with logit constraints

Please note that I have a limited knowledge of nonlinear programming (but I have taken linear programming), and part of my intention is to get readable references on this type of problems (short of ...
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Constrained maximin problem

Let i) $\mu = [\mu_1,\mu_2,\mu_3]\in\mathbb{R}^3$, such that $\mu_2 > \mu_1$, $\mu_2 >\mu_3$ fixed, ii) $\lambda = [\lambda_1,\lambda_2,\lambda_3] \in \mathbb{R}^3$ such that $\lambda_1 \geq \...
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Minimizing matrix cost function

I have a very basic knowledge of minimization problems mainly limited to curve fitting. I need help to understand how to minimize the following cost function with respect to $X$. $$||Y-XD^T||^2_2+\...
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Lagrange multiplier for non-convex optimization

An object is free to move within the set $S=$ { $(x, y) \in R^{2} | x+y \leq \cos (y-x) $ }. It tries to come as close as possible to the point $\mathrm{P}=(1,1) > .$ Where should it go? In ...
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How to use the KKT-conditions for a not-differentiable function using subdifferentials.

First some notation. Let $\dfrac{\partial}{\partial \textbf{x}} f(\textbf{x})$ determine the gradient for a funcion $f:\textbf{R}^n \rightarrow \textbf{R}$, and let the subdifferential be determined ...
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Projection of an interior point of an ellipsoid onto itself

Consider $E:=\{x \, | \, x^TDx=1\}$, an ellipsoid constructed by the daigonal matrix $D=\text{diag}(d_1,d_2,\dots, d_n)$ with $d_i>0,\ \forall i\in [n]$. Suppose that $z^TDz=\alpha<1$, so $z$ is ...
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How to discretize nonlinear Poisson Equation

I am trying to solve a nonlinear poisson equation of the form: $u_{xx} + f(u_y)u_{yy} = 0$. I would like to use Newton's method to handle the nonlinearity, however I am not sure when/how to ...
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Adding robustness to the objective function of the facility location problem

Let us first define a simplified facility location problem as follows: $$\min \sum_{i=1}^{n}\sum_{j=1}^{m}d(i,j)x_{i.j}$$ subjected to: $$\sum_{i=1}^{n} x_{i,j}\geq1, \forall j$$ (Every customer ...
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CPLEX output after linearization

I have a quadratic objective function that is a result of multiplying two 0-1 decision variables $X$ and $Y$. I solved it using LINGO and CPLEX. I noticed that there if I introduced a new variable $Z=...
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Coervive Map on a Banach Space

If $B$ is a finite-dimensional Banach space then norm-coercivity and coercivity coincide since the weak and strong topologies coincide. However, if $B$ is infinite-dimensional, say $B=L^p$ for $p\in [...
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Solving Max-Min problem by first doing min and then max

Assume that we have the following max-min optimization prblem \begin{align} \max_i \min_x f_i(x),\quad i\in\{1,2,\cdots,I\} \end{align} where the optimization problem $\min_x f_i(x)$ is solvable and ...
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Convexity of quadratic form with respect to new set of parameters

I understand that the quadratic form, $f(x) = x^TAx$, is convex with respect to $x$ so long as the matrix $A$ is positive semi-definite. If we assume that $A$ remains positive semi-definite, is the ...
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Euler Lagrange equation with initial velocity constraint

Let $q^* \geq 0$, and $L \in{} C^1(\mathbb{R}^3)$. I want to find a function $P \in{} C^1[0,q^*]$ that maximizes the functional $I(P):=\int_0^{q^*} L(q,P(q),P'(q))dq$ subject to the restrictions $P(0)=...
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Knapsack problem on 2D or 3D space

Considering a series of rectangle items with known size $(a_1,b_1),(a_2,b_2)\cdots,(a_n,b_n)$, and a big rectangle box with size $(A,B)$ Question 1: How to fill the box with the items that minimize ...
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Optimality conditions for quadratic programs

Here is the problem I am currently working on: I am really struggling with 4b and these more general quadratic programs. I've tried applying Wolfe's method for Quadratic Programs, but that isn't ...
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Show that if this nonbinding constraint is deleted, it is possible that $\bar{x}$ is not even a local minimum

Hello guys I am looking for some help for this nonlinear problem Let $\bar{x}$ be an optimal solution to the problem of minimizing $f(x)$ subject to $g_{i}(x)\leq0, i=1,...,m$ and $h_{i}(x)=0, i=1,.....
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Optimization of quadri-linear system

I have run into a problem where I need to optimize (either minimize or maximize) - the following: Given a symmetric positive definite $n \times n$ matrix $\Sigma$, to find an n-vector $x$, such that $...
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Maximize $\log(2)+\log(3/2)x+\log(2)y+\log(5/2)z$ if $x+y+z\leq 1$ and $(y+z)^2+2x-x^2-2xy\leq 1-2\gamma$, $0.24 \leq \gamma \leq 0.25$

I am trying to maximize the function $$f(x,y,z)=\log(2)+\log(3/2)x+\log(2)y+\log(5/2)z$$ with the following constraints: $$x\geq 0, y\geq 0, z \geq 0,$$ $$x+y+z\leq 1,$$ $$x+y\geq 4/5,$$ $$(y+z)^2+2x-...
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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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6answers
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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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1answer
18 views

Geometric Programming With Reciprocal Objective Function

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=...
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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-...
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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 \\...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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“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 $\hat X = A^\dagger Y$...
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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 \\...
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1answer
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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 ...
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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 ...
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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,...
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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_{...
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Bounding quadratic objective maximized over probability simplex

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{...
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1answer
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Dual of a function in closed form.

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...

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