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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 ...

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Obtaining a valid subgradient

I have a nonconvex function, which could be non-smooth in some places. I also have the expression for computing the gradient vector of the function. This vector is valid as a gradient wherever the ...
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When is LICQ useful in KKT conditions?

KKT establishes a set of criteria for differentiable optimisation problems related to strong duality (i.e. when primal optimal equals dual optimal). In particular, KKT conditions are necessary for ...
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Question about strong convexity

I don't really know how to begin. I tried substituting $y$ for $x + h$ and taking the Taylor approx. of $f(x + h)$ around $f(x)$. The RHS becomes $h^T \nabla f(x) + \phi(x)$ Where $\phi$ is our ...
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Do multivariate Householder methods exist?

Newton's method can be extended to higher-order versions using Householder's method. Newton's method can also be extended to the case of multivariate inputs, sometimes called the "Newton-Raphson ...
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Find the point that is at distance $1$ from $(0,0,0)$ and at distance $3$ from $(1,2,3)$ that is closest to $(5,-2,4)$.

I have this question : Find the point that is at distance $1$ from $(0,0,0)$ and at distance $3$ from $(1,2,3)$ that is closest to $(5,-2,4)$. Here is my failed attempt. I used Lagrange ...
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Is dynamic programming suitable for a specific optimization problem?

Let $c,\,\mathcal{P}_0,\,\mathcal{P}_1,\,\mathcal{P}_2,\ldots$ be a sequence of positive real numbers. Let $N\in\{1,\,2,\,3,\ldots\}$ and let $t\in\{0,\,1,\,2,\ldots\}$, with $N$ and $t$ fixed. ...
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What are the properties of the solution of a nonlinear programming with linear objective function

In a constrained nonlinear programming with a linear objective function, are there any results/properties of the solution? The constraints are nonlinear nonconvex.
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What can be the convex relaxation of a quadratic matrix inequality?

I am trying to relax the Quadratic Matrix Inequality given as: $$W \leq X^TX+Y^TY \\ W\geq 0 $$ Here, $X,Y,W \in \mathbb{R}^{n\times n}$ matrices. These two are to be solved along with one linear ...
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Lagrange Mutiplier for inequality constraint

I'm a bit confused about Lagrange multipliers. I know it works wonders if I only have equality constraints. Whenever I have inequality constraints, or both, I use Kuhn-Tucker conditions and it does ...
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28 views

Conditions for $\Vert f \Vert_2^2$ to be convex

Question: I am currently looking for general conditions for the function $\Vert f \Vert_2^2$ to be convex, where $f:C \to C$ and $C \subset \mathbb{R}^n$ is compact. References on this problem or ...
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Do redundant constraints help in big-M reformulation?

I am trying to reformulate an optimisation problem with unknown $x$ of dimension $K\times 1$ into a mixed-integer program using big-M transformation. In this respect, among my constraints, I have ...
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(Generalized) Benders Decomposition of MINLP Leads to Linear Master and Sub Problem

I have a mixed-integer nonlinear program (MINLP) which I want to apply (Generalized) Benders Decomposition (GBD) to. The nonlinearities exclusively originate from products of first-stage decision ...
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Is $f(x,y) = x\log(x)+y\log(y)$ a coercive function?

From Peressini, Sullivan, Uhl, the mathematics of nonlinear programming, A function is coercive if $\lim\limits_{\|x\| \to \infty} f(x) \to \infty$ and super-coercive if, $\lim\limits_{\|x\| \to \...
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Quasi-Newton Methods no-change Condition Requirement

In standard quasi-newton methods for fixed point iteration, it looks there is two required conditions. The first one is secant condition: $$J_{k+1} \Delta x_{k} = \Delta f_k $$ where $\Delta f_k = f(...
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Nonlinear optimization of a matrix with the costraint to be orthonormal

I'm trying to find the matrix x which minimize the following cost function : $J =||B_b -x*B_n||^2$ with the constraint that x has to be an orthonormal matrix. I'm trying to use MATLAB fmincon tool, ...
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Local minima for the quadratic penalty method

Let $x\in\mathbb{R^n}$ be a strict local minima for the problem \begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} && f(x)\\ & \text{subject to} && h(x)=0 ...
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Is there something like “Integer non-linear programming” without mixed-integers

I just wanted to know whether problems exist that belong to the category "Integer non-linear programming" without belonging to the category "Mixed-integer non-linear programming"? While having ...
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How to solve a non-convex programming problem?

Let $A$ and $C\ $ be $n\times n$ symmetric matrices, and $A\bullet C=Tr(A^TC)$. Let $S\subseteq [n]\times [n]\times [n]$. Define a non-convex programming problem as follows. \begin{equation} \begin{...
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Interior Solution to $\max( ax^z+by^t)$

I am trying to understand why the interior solution to this problem doesn't exist. The only solution is $0$ or $1$, is this correct? I actually find the opposite: the boundary solutions do not exist. ...
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Non-linear constrained problem transformation to equivalent un-constrained problem

I have the following non-linear optimization problem: min $f(x, y, z) = x + y + z$ s.t. $x^2 + y = 3$ $x + 3y + 2z = 7$ Is there a way to transform this problem to an equivalent minimisation ...
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How to find the Fenchel dual of this expectation

I'm trying to figure out the Fenchel dual of the following expectation minimization problem: $$ \min_f E[l(x,f(z))] $$ where $x\sim S$ and $z\sim T$ ($S$ and $T$ are two distributions here). I'm ...
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The optimum of a function based on directional optimals

In a problem of minimizing a (linear funtion) under (nonlinear) constraints, I found that a solution $x*$ solves the problem in each of the directions. Can I say that $x*$ solves the problem in a ...
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Eigenvalue analysis, subject to boundary conditions

For some non-linear finite element program I have a Tangent stiffness matrix $\textbf{K} \in \mathbb{R}^{n\times n}$, which is symmetric. I want to find the Eigenvector corresponding to the smallest ...
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(Worst and average case) Arithmetic and time complexity of a quadratic programming problem by interior point method

My understanding of complexity measures are at basic level. So, please excuse me for asking basic question, if it sounds like. I am trying to understand the arithmetic and time complexity of a ...
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Constrained Optimization problem to unconstrained problem

I have a constrained optimization problem, I would like to convert this constrained problem to an unconstrained problem, specifically the constrained problem is constrained on convex set, which is: $$ ...
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The meaning of active and non-active constraints

Consider the constrained minimization problem min $f(x), x \in \mathbb{R^n}$ s.t $h_i(x)=0, i=1,2,...m$ $g_i(x) \leq 0 , i=1,2,..k$ Now the author states: " For a feasible solution $x$, some of ...
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How to linearize the product of a binary variable X and a continuous variable Y

I have equations and I want to linearization them they are c>=(st+pr)x where st is continuous variable , pr is continuous variable and x is binary variable. c=>st(x+z) where st is continuous ...
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What are the conditions for a coordinatewise optimum to be a local optimum

if $x*$ is a coordinatewise optimum of a function $f$, (i.e $x*$ is an optimum of $f$ in all directions), what are the conditions (on the function $f$ or $x*$) for $x*$ to be a local optimum of $f$.
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If x* is a local optimum of a function in all directions, could it be optimal in a neighborhood of x*?

In a Euclidean space whose base is denoted $(e_1,e_2,...e_N)$. suppose that $x*$ is the local minimum of a function $f$ in all direction. Could we say that $x*$ is an optimum of $f$ in a neighborhood ...
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Lagrangian: $\min_{\mathbf{X} \in \mathbb{R}^{N \times K}} \left\|\mathbf{Y}-\mathbf{X}\right\|_F^2$ s.t. $\mathbf{A}\mathbf{X} = \mathbf{B}$

Let us say that the optimization problem can be posed in the matrix form as given below $\min_{\mathbf{X} \in \mathbb{R}^{N \times K}} \left\|\mathbf{Y}-\mathbf{X}\right\|_F^2$ s.t. $\mathbf{A}\...
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Newton's method - optimal step size

I am wondering how to prove the following affirmation: We have : Min $f(x) = \frac{1}{2}x^TDx - c^Tx$ with $f: R^n → R^1$ and $D$ a symetric positive matrix of size $nxn$. Suppose $d_k = -\...
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How to minimize $L_1$ norm in a subspace when $L_2$ norm is fixed?

I want to solve the following optimization problem $$\text{minimize}~~\lVert \vec{x} \rVert_1$$ $$\text{subject to}~~\lVert \vec{x} \rVert_2 = 1$$ $$\text{and}~~A \vec{x} = \vec{0}$$ where $\vec{x} \...
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Squashing function Capsule Networks

I want to understand how the squashing function of the capsule network works? Like, in the image, I want to know what's the application of unit scaling? Please let me know. enter image description ...
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Is it possible to turn this into a (standard) integer convex knapsack problem?

I have found a solution algorithm for integer knapsack problems of the following form: $\max\limits_{x_j \in [l_j,u_j]} \sum_{j=1}^n f_j(x_j)$ such that $\sum_{j=1}^n g_j(x_j) \leq b$ ...
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A class of non-convex functions

What are some examples of not necessarily convex functions mapping $\mathbb{R}^n \rightarrow \mathbb{R}$ which are smooth, have gradients of bounded norm and have a global/local minima? I would be ...
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How do you take the derivative for a line search with respect to the step length?

I'm reading about line search methods in Numerical Optimization by Nocedal and Wright and am wondering how the derivative of the line search function is obtained. In exact line search methods with a ...
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MIQP problem slow to solve: how to rewrite it?

I am looking for suggestions on how to rewrite a MIQP problem. Let me firstly introduce the problem Notation: The unknown vector is $x$ with size $(4*2+225*2)\times 1$. We can think of the ...
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What is the logarithmic transformation of this optimization problem?

Given $$n, C, r_i, p_i,a_i \quad∀ i={1,2,...,n} $$ I want to solve this optimization problem: $$\text{maximize} \quad f(x_1,x_2,...,x_n)=\prod_{i=1}^n {{(x_i-a_i)}^{p_i}} $$ $$\text{s.t.} \quad {(x_i/...
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What are the directions of research in Numerical Optimization?

I have just begun reading in the field of Numerical Optimization. Are people trying to invent new Algorithms? or proving the convergence of Heuristic Algorithms? and what else? What are the tools a ...
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Can KKT be used in minimization s.t to constant param

Can KKT be used : min g(x) s.t x>=constant where constant > 0 I have read this The Kuhn-Tucker method: here says that This is an alternative, and ...
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Question on Probability maximization

Mr A wants to join a Gamer's club. There are two identical boxes filled with Red and Green balls, and he has to pick up a green ball in order to join the club. You are required to allocate balls in ...
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Maximise $z = \frac{y}{2x+2y}+\frac{50-y}{200-2x-2y}$ given that $x+y$ is non zero and $x+y<100$. Also, $x\leq50$ and $y\leq50$ and non-negative.

Z is actually a probability function. I am finding where the probability is maximized. But I could find no way how to maximize this function. Original question is as follows: Mr A wants to join a ...
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Solving a non-linear program $\min x + y^p$

Can anyone tell me how I can solve the following NLP for fixed p > 1: $$ \min x + y^p \\ st \ x+y=1 \\ \ x,y \ge 0 $$ Thanks! I tried using KKT theorem, but it seems this program has no Slater ...
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1answer
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Bounding using Kantorovich inequality

In my homework, I've made it to the point where I obtained this value $$\frac{(v^Tv)^2}{v^TQvv^TQ^{-1}v}$$ where $v \in \mathbb{R}^n$ and $Q\in \mathbb{R}^{n\times n}$ and $Q$ is symmetric positive ...
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How to solve this Non linear optimization problem?

I need to minimize the below-mentioned expression. $ L = min (a_0-b_0*(p_1+p_2))^2 + (p_1*y1+p_2*y)$ ,with s.t p_1 >=0 ,p_2 >=0 Here ...
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$\min f\ \mbox{s.t. $Ax=b$}$. If $\nabla f(\overline{x}) = A^t\lambda, \nabla^2 \overline{x}\ge 0$, prove $\overline{x}$ is local minimizer

Let $f:\mathbb{R}^n\to\mathbb{R}, f\in C^2$. Let $\overline{x}\in\mathbb{R}^n$ such that $A\overline{x}=b$ and such that there exists $\lambda\in\mathbb{R}^m$ with $\nabla f(\overline{x}) = A^t\...
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Equivalence of two Lagrangian Optimization Problems with equal restrictions

Assume we have given two strictly convex functions $f_1$ and $f_2$ that map from $\mathbb{R}^p$ to $\mathbb{R}$. We want to find the vector $X\in \mathbb{R}^p$ that optimizes them. Both functions ...
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Proving complicated transcendental inequality

Suppose we have a function $f$ of four posirive real numbers $a,b,c$ and $d$ in a domain that, for a given real number $0<r<1$ they satisfy $$rc<b<a,$$ $$rc<rd<a.$$ We then have $$...
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60 views

What is minimum distance between $y = (y_1,y_2)$ and the curve $f(x) = 1/x$?

This question is a follow up on a question (Show that $\{ x \in \mathbb R^2 : x_1x_2 = 1 \}$ is closed) that I asked earlier. The question is given by the following. Given a point $\boldsymbol y = (...
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1answer
61 views

$\min \frac{1}{2}x^tBx + c^tx$ subject to $Ax = b$ has unique solution $\iff$ $z^tBz>0$ for all $z\in \ker A, z\neq 0$

Consider the problem $P$ of minimizing $\frac{1}{2}x^tBx + c^tx$ subject to $Ax=b$, where $\{x\in\mathbb{R}^{n}|Ax=b\}$ is non empty and $B$ is symmetric. a) Prove that if $P$ has solution, ...