Questions tagged [nonlinear-programming]

Questions about the theory and numerical algorithms for optimizing (minimizing or maximizing) nonlinear functions, possibly subject to equality and/or inequality constraints.

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70 views

Large-scale optimization of nonlinear equations

I'm looking to find a computationally efficient solution to a large system of nonlinear equations. I'm trying to maximize the following function: $$ f(\vec{x}) = \sum_i^N C_i (x_i-A_i)x_i^{\epsilon_{...
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0answers
30 views

Method for implementing QP solver with matrix terms?

I am trying to implement (in code) a QP solver for the following equation: $$\min_{u} u^{T} Wu$$ $$s.t. \; \beta u = \tau_{ref}$$ $$ Au \leq b $$ See this document, section 5.1 (Page 35) $u$ is a ...
3
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2answers
120 views

MINLP with GEKKO - Modeling discrete variables

I'm trying to define a MINLP optimization problem with GEKKO in Python, and I want to use some variables with fixed values. For my first variable, x1, I need to define the following values (as would ...
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1answer
87 views

Preconditionning for solving a non-linear system of equations with least squares

I am trying to solve a large system of non-linear equations (about a few hundred equations and variable but with less variable than equations). Given that the system is really sparse and large I am ...
3
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0answers
27 views

How to set up and solve acceleration-limited trajectory optimization problems?

I've been trying to learn how to solve simple acceleration-limited trajectory planning problems. I'm working in C++ and I've been using the Eigen library to do linear systems solving. I'm doing the ...
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1answer
53 views

Research articles on MultiObjective Non-Linear Programming (MONLP)

I'm looking for papers dealing with multi-objective non-linear programming which could help me implement an algorithm to solve my problem. My problem is : Maximize $f(x) = c \cdot x$, while ...
3
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1answer
74 views

Geometric Programming - symbolic version

I am interested in finding minimizers of functionals of the type $\sum x^ay^bz^c$ where the exponents are 1, 0 or -1. I have codes to find such minimizers when they exist up to machine precision, ...
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0answers
46 views

Finite dimensional optimization problem over dynamical system

I am interested in solving numerically the following mathematical problem Consider an ode of the form $$ \dot q(t) = f(q(t),t_1,\ldots, t_N),\qquad t\in [0,T], $$ where $q\in \mathbb{R}^n$ is the ...
2
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1answer
76 views

Nonlinear conjugate gradient with orthogonality constraint

I have to solve a set of nonlinear optimization problems in the subspace defined as the orthogonal space to a given vector. More precisely, $$ \arg\min f(\vec x) \qquad \text{with} \qquad \vec x \...
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1answer
122 views

Correct way to model an embedded reinforcement (non linear FEM)?

I need to add to an existing FEM solver some embedded reinforcement element. This would give me the possibility to model/solve concrete structure (reinforced with steel rebar) taking into account the ...
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0answers
97 views

Connection between piecewise linear basis functions and RELU activation function

ReLU activation is defined as follows $$\sigma(x)=\max(0, x).$$ Let's assume that I have deep network of 1 hidden layer, than output from my layer has form $$ f(x)= \sigma(Wx +b), $$ where matrix W ...
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1answer
42 views

Slightly change two vectors to satisfy a constraint

$\vec{a}\cdot\vec{b} \approx c$ $\vec{\alpha} \cdot \vec{\beta} = c$ $\vec{\alpha}$ is close to $\vec{a}$ and $\vec{\beta}$ is close to $\vec{b}$ Given $\vec{a}$, $\vec{b}$ and c, how to find $\vec{\...
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1answer
216 views

Why does Newton's method with Linear Equality Constraints use KKT condition?

Goal: Optimize convex function $f(\vec{x})$ subjected to constraint $A\vec{x} = \vec{b}$ starting at a point $\vec{x}_0$ that satisfies the constraint. The problem only has equality constraint. Why ...
1
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1answer
226 views

Piecewise-Linear Quadratic Optimization for an “Almost Convex” Problem

I have a 7-14 dimensional piecewise linear cost function I'd like to minimize with two quadratic terms of the form: $$ f(X) = X^tCX + d \sum_i |x_i-x^*_i|^2 + \sum_i P_i(x_i-x^0_i) $$ $$ \sum_i x_i ...
5
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2answers
1k views

Scaling of optimisation function in non-linear least squares problem

Suppose I am minimizing the following function: $$g(\alpha)=\sum_{i=1}^n(y_i-f(x_i,\alpha))^2,$$ where $y_i$ and $x_i$ are data, $f$ is a known non-linear function and $\alpha$ parameter (of ...
2
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0answers
270 views

Convergence of a very large non-linear least squares optimization

(note: I also posted this question on stackoverflow before finding this community here, which seems a better place for it) I'm trying to solve the following problem: I have a lot (~80000) surface ...
4
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1answer
195 views

How to solve the following Frobenius norm-minimization problem?

Background We know how to solve the following minimization problem $$ \min_{X} \lVert AX - B \rVert_F^2 $$ But what about the extended version? $$ \min_{X} \lVert A \begin{bmatrix} X & X^2 \...
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0answers
69 views

Nonlinear least square optimization

Problem description Given data at many time instance $t$, $$\min _{\alpha, \Lambda, \beta} \lVert y(t) - \alpha e^{\Lambda t} \beta \rVert_F$$ with $$ \lVert \alpha \rVert_2^F = 1 $$ where $y(t) \...
81
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17answers
90k views

Is there a high quality nonlinear programming solver for Python?

I have several challenging non-convex global optimization problems to solve. Currently I use MATLAB's Optimization Toolbox (specifically, fmincon() with algorithm=<...
2
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1answer
31 views

Constraints 'exactly/at most one non-zero element' without binary variables

In a much larger MINLP problem, I have set of variables $\{a_{ij}\}_{m,n}$, such that $0 \leq a_{ij} \leq 1 $ for all $i$, $j$, which I could think of as a matrix, for which I have two requirements: ...
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2answers
663 views

GAMS solvers: which one to use

The other day I had a discussion with a friend about the GAMS solvers and we were wondering what are the mathematical differences between the solvers. Which one to use for which kind of problem? How ...
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2answers
2k views

Optimization with matrix determinant as constraint

I'm solving a constrained optimization for matrix $\mathbf{A}$ with dimension 6x6, where one of the constraints is $\mathrm{det}(\mathbf{A})>0$. I use the NLopt package to solve my problem and ...
0
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1answer
57 views

reduced system: primal-dual interior point method for nonconvex constrained problem

When solving a reduced KKT system of a nonlinear (and nonconvex) constrained program after eliminating slack and dual variables, how do we actually take the next step in a primal-dual method? For ...
3
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1answer
127 views

Obtaining a feasible solution for underdetermined system of linear equations satisfying inequality constraints

I would like to obtain a feasible solution for an under-determined system of linear equations, $$Ax=b$$ where, $A \in \mathbb{R}^{7\times9}, \, x \in \mathbb{R}^{9\times1}\text{and } b\in\mathbb{R}^...
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0answers
52 views

Nonlinear Sylvester-Like Equation

Maybe you can point me to some results already developed for this. I have to solve for $X$ the following "Sylvester-like" equation: $$ AX - XB = F(X)$$ where $A\in\mathbb{R}^{a\times n}$, $B\in\...
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2answers
89 views

How can i solve this non-convex multi-variable optimization problem?

I want to solve the following optimization problem: $$\min_{A,B,X} \|Y-AX\|_F^2 + \lambda_1 \|Z-BX\|_F^2+ \lambda_2 \|B\|_F^2$$ $$s.t ~~x_{ij}~ \geq 0$$ in which, $Y$ and $Z$ are data matrices and ...
1
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1answer
587 views

Direct multiple shooting (numerical optimal control)

Please, I am currently implementing direct multiple shooting methods* and I need one simple but fundamental concept answered: When I want to provide not only objective function value (the result of ...
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0answers
72 views

Nonlinear global optimization algorithm that can use dynamic programming

I've asked this question on stackoverflow 2 weeks ago, but, judging by zero response, that probably was the wrong forum. Therefore copying it here: Let F0,...,Fn ...
3
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1answer
231 views

Why do active set methods or the simplex method pivot only one variable at a time?

Why do active set methods or the simplex method pivot only one variable at a time? Ostensibly, we could add multiple columns to the basis during pivoting, but the standard presentation of the methods ...
0
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1answer
64 views

Why can't we discretize continuous domains in distributed non-convex constraint optimization problems?

Consider a non-convex distributed optimization problem. We have $X$ = a set of $n$ decision variables: $x_i$ where $i=1..n$ and $x_i \in R$, the set of Reals. We have $F$ = a set of $m$ constraint ...
3
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1answer
259 views

Symmetric nonnegative matrix factorization

Suppose $A\in\mathbb{R}_+^{n\times n}$ is symmetric. I would like to factorize $A\approx UU^\top$ by solving $$ \begin{array}{rl} \min_U & \sum_{ij} \left(A_{ij}\ln\frac{A_{ij}}{[UU^\top]_{ij}}+[...
5
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1answer
89 views

$L_2$ projection with integer constraints and prescribed sum

Suppose I am given a vector $v^0\in\mathbb{R}^n$ and integers $k,\ell\in\mathbb{Z}$. Assuming $k$ is close to zero (e.g. $0\leq k\leq5$), is there an algorithm for solving the following integer ...
1
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0answers
56 views

Object-oriented non-linear solving in python

I'd like to build a system in Python, consisting of (broadly speaking) objects which are internally described with (not necessarily just linear) equations, that I can connect with each other - similar ...
1
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0answers
68 views

Minimizing the products of variables

My problem Maximize $$\min_{i} \{\ c_i \cdot \prod_{j \in A(i)} {x_{j}} \prod_{j \in B(i)} {y_{j}} \} $$ Subject to \begin{align} &\sum_{j \in C(k)} x_{j} = 1,\ \forall k \\ &l \leq x_{j}...
2
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1answer
231 views

Optimization of multiple functions

I have 3 functions which consist of 6 variables $p_1,p_2,p_3,p_4,p_5,p_6$. The value of each function is equal to $x$ (say): \begin{align} f_1 &= \operatorname{sign}(2-p_1) \sqrt{|2-p_1|} + \...
4
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2answers
788 views

nonlinear conjugate gradient for multivariable functions

For the optimization problem $\underset{\mathbf{x}\in \mathbb{R}^n}{\operatorname{argmin}} f(\mathbf{x})$, we can use the following standard nonlinear conjugate gradient method to find the solution: $...
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0answers
209 views

Constrained optimization: Stationary point vs. Nash point

1s question: definition of stationary point for constrained optimization As far as I know, a stationary point of a constrained optimization problem is a stationary point of the Lagrangian (that has ...
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0answers
47 views

How is the Gastner-Newman equation implemented to create value-by-area cartograms?

There is a paper called "Density-equalizing map projections: Diffusion-based algorithm and applications" by Michael T. Gastner and M. E. J. Newman, which explains their algorithm (which is based in ...
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1answer
279 views

Example Problem to Demonstrate BiCGStab

So our team has been able to code up a BiCGStab implementation for a class project, and we'd like a potential example problem to try it out on. So far, we've talked about a 1D Laplacian with Neumann ...
5
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1answer
135 views

Largest hypercuboid inside a polyhedron

Given a polyhedron $\mathbf{Ax} \leq \mathbf{b}$, how to find the largest hypercuboid, with unknown center $\mathbf{x_{0}}$ and side lengths $2\epsilon_{i}$, which are aligned along the co-ordinate ...
0
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1answer
5k views

Backtracking-Armijo Line Search Algorithm

EDIT based on comments below: I add the mathematical formulation of my problem below. I am trying to solve an equation of the form $$ \partial_t f(x,y,t)= (\partial^2_x +\partial^2_y) f(x,y,t) \equiv ...
1
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2answers
43 views

Optimisation of purely integer quantity with bound-constraints for a 1D expensive function whose analytical form is not available

I have a computationally expensive objective function, whose analytical form is not available. The only input argument to the objective function is an integer variable. The goal is to compute the ...
3
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2answers
66 views

Gradient of function after renormalization of variables

I have to minimize a function $f(\mathbf{x})$, where the vector $\mathbf{x}\in\mathbb{R}^n$ satisfies $|\mathbf{x}|=1$. So I tweaked the code of $f$ so that it renormalizes $\mathbf{x}$ as the first ...
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1answer
87 views

Perturbation in bounds given the perturbation to constraints

Given a feasibility problem with both inequality and equality constraints, I'm interested in the sensitivity of the bounds of the region to changes in the constraints. To help with answering the ...
2
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0answers
64 views

Global optimization with known distributions of some variables

I'm solving simple single-objective multidimensional global optimization problem using various stochastic algorithms like Monte-Carlo, GA and other evolutionary approaches. The task is formulated as ...
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0answers
93 views

2D Multilateration with constraints as distance information

In my current problem, I am looking for an algorithm to reconstruct the position of multiple points in the 2D euclidean plane with incomplete distance information, $d_{ij}=||x_j-x_i||\in\text{dom}_a$. ...
1
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1answer
110 views

SOCP: Recovering primal from dual

Consider the following second-order cone program (SOCP): $$ \begin{array}{rl} \min_x & c^\top x\\ \mathrm{s.t.} & \|A_ix+b_i\|_2 \leq c_i^\top x+d_i \ \forall i \end{array} $$ Suppose I solve ...
2
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1answer
210 views

Minimizing linear objective on intersection of convex sets

Suppose I wish to solve the following optimization problem: $$ \begin{array}{rl} \min_{\mu\in\mathbb{R}^n} &\mu^\top c\\ \textrm{subject to} & \mu\in C_1\cap C_2\cap\cdots\cap C_k, \end{array} ...
2
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0answers
208 views

Using Line Search Method for Constrained Optimization

Suppose we have a $f(x)$ to be minimized (we only know that $f(x)$ is three-differentiable), and a feasible, convex set of $S$ such that all $x$ belong to $S$. Using line search method, how we can "...