Questions tagged [nonlinear-programming]
For questions about mathematical optimization problems involving a nonlinear objective function and/or nonlinear constraints.
138
questions
2
votes
2
answers
110
views
Difference between constraint formulation and performance
I am wondering about the characteristics and performance of some constraints with only binary variables. I assume that solving (integer) linear programs is faster than quadratic ones.
At first:
$$
a,b,...
- 61
2
votes
0
answers
74
views
Tolerance in Baron
I am solving a relatively large NLP (feasibility) problem using the solver Baron. I need the optimal solution to satisfy the constraints within a tolerance of 1e-15. However, it seems that, unlike ...
- 111
5
votes
2
answers
97
views
Analyzing the output of IPOPT
I am solving a feasibility (No objective) problem in IPOPT. I got the following output:
I see that the violation of the constraints are of order 1e-15. What is the meaning of dual infeasibility 1e-07 ...
- 111
1
vote
0
answers
43
views
Dual of a quadratic constraint
This is my model.
\begin{align}
\min_x&\quad\sum_{e\in E} X_e p_e \\
\text{s.t.}&\quad\sum_{e \in E: T(e)=i} X_e - \sum_{e \in E: H(e)=i} X_e = \begin{cases}1, \;\text{if}\;i=s\\-1,\;\text{if}...
- 353
4
votes
2
answers
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How can we write a binary variable as a power to a constant number?
Let $x_{i,j}$ be a two-dimensional binary variable.
Is it possible to write $x_{i,j}$ as a power to a number?
For example:
$$1- 0.3^{x_{i,j}} $$
- 307
1
vote
1
answer
79
views
If $x=\min\{f(\mathbf{a}),1-\epsilon\}$, how can we model and partition $x$?
I have been dealing with a problem for sometime and although tried different things and asked some questions before, I think the problem might be somewhere that we didn't look before.
Variables $0\le ...
user9659
2
votes
1
answer
61
views
Linearizing $y=\sum_{i=1}^n(z+c)\left(\frac{r_i^2}{1-r_i}\right)\phi_i$
Variables $0\le x< 1$, $y,z\ge 0$. We have a constraint
$$y=(z+c)\frac{x^2}{1-x},$$
where constant $c>0$.
We partitioned $x$ into $n$ intervals of equal length and defined a new variable $\phi_i=...
user9659
3
votes
1
answer
60
views
Methods to solve integer linear inequalities with products of two variables
I'm interested in solving the following system of equations over the integers:
\begin{align*}
x_l^3 &\le x_l^1x_l^2 & \text{ for } l = 1,\ldots,s \\
A x &\le b \\
0 &\le x
\end{align*}
...
- 131
2
votes
1
answer
97
views
Passing exact number of allocations as constraint to pyomo in a sourcing problem
I am solving a sourcing allocation optimization problem. Here I have let's say two brands. Each brand has a raw material demand across the 3 plants (Demand in kg)
Brand 1
Brand2
Plant 1
3000
2000
...
2
votes
0
answers
128
views
Solving minimax problems with Gurobi
I want to solve a problem of the form $\min_x\max_y f(x,y)$ using Gurobi, where $x,y\in [0,1]$.
Is there a simple way to model this in Gurobi? I've seen examples where the domain of $y$ is finite, but ...
- 21
4
votes
1
answer
55
views
Can we reformulate a min-max nonlinear integer programming problem with the same optimal solution?
I'm trying to solve the below min-max nonlinear integer programming problem with the following objective functions:
$$\min_{x_i,y_i} \max_i \left\{\frac{x_i}{2+3y_i} + \frac{2x_i}{4+7y_i}\right\} \\ \...
2
votes
1
answer
153
views
Solver for nonlinear semidefinite optimization
Totally new to optimization. Is there an easy-to-use solver, package, (free) software for solving nonlinear semidefinite optimization problems?
- 129
7
votes
4
answers
814
views
is prime? in Operations Research
Is there a way to linearize is prime? in Operations Research?
is prime(n) being true if $n$ is a prime number or false otherwise....
- 533
4
votes
0
answers
36
views
Does knowing the "correct multipliers" for globally optimal first-order critical points help you algorithmically?
Consider the following nonlinear optimization problem:
\begin{align*}
&\min f(x) \\
\text{such that } &h_1(x) = 0, \\
&h_2(x) = 0, \\
& \vdots \\
& h_m(x) = 0,
\end{align*}
where $...
- 41
0
votes
1
answer
135
views
Algorithms and analytical solution methods/tools to solve an optimization problem for a particular objective function
I have the following objective function:
\begin{equation}
\min_{I_{i,v}} \ \sum^{N_v}_{v}\sum^{TT_v}_{i} \ C_{loss,cyc}
\end{equation}
where $I_{i,v}$ is the only positive decision variable. $...
- 327
5
votes
1
answer
94
views
How to handle a non-separable bilinear objective function in the special case of decoupled constraints?
I have a large number of (10000+) non-negative, real decision variables $x_i$ and $y_j$.
Let $I$ and $J$ be the index sets associated with $x$ and $y$, respectively.
Let $\bar{I}$ and $\bar{J}$ be non-...
- 135
5
votes
1
answer
165
views
How to handle a bilinear objective function in the special case of decoupled constraints?
I have decision variables $x_i$ and $y_j$, real positive variables.
I would like to minimize objective function
\begin{aligned}
\min \quad & \sum_{ij} x_iy_j \\
\end{aligned}
All constraints are ...
- 135
1
vote
0
answers
167
views
In DOcplex, how can I get value of variable for objective function before solved?
In terms of python based atypical math model as shown below, which is unlike nonlinear or quadratic form in my opinion, I have difficulty in accessing value of variable before optimization as ...
2
votes
1
answer
242
views
How can I linearize this nonlinear variable relationship?
Assume a mathematical optimization problem with two positive continuous variables:
0 <= x <= 1
0 <= y <= 1000
I am seeking an efficient way to express ...
- 29
5
votes
2
answers
152
views
Linearizing $x^2/(1-x)$ by partitioning the interval $0<x\le X$
We have two decision variables
\begin{align}
& 0<x\le X,\\
& 0<y\le Y,
\end{align}
where both $X$ and $Y$ are two sensible upper bounds on our decision variables.
We also have a ...
user9659
3
votes
0
answers
69
views
Linear program with an additional second-degree utility term
I would like to solve a problem obtained from a LP by adding a second degree term to its utility.
A simple example would be the following (with $c_i > 0$):
$$ \min xy - c_1 z_1 - c_2 z_2
\\ \...
- 31
3
votes
1
answer
197
views
Linearizing division of two variables
For all $i \in I,j\in J$ and $k\in K$, define variables $x_{ij}, z_{ijk}\in\{0,1\}$, $y_{ij}\geq 0$ and constants $c_j, e_{ijk}, d_j, f_j >0$.
We have the following constraint
$$\sum_{j\in J_1}c_j\...
- 103
4
votes
1
answer
217
views
Can we simplify (perhaps linearize) this constraint?
We are dealing with a stochastic model and one of the constraints is
\begin{align}
y_j=\frac{\sum_{i \in I}\sum_{k \in K}\mathbb{E}\left[X_{ik}^2\right]x^k_{ij}}{\sum_{i \in I} \sum_{k \in K} \mathbb{...
user9659
0
votes
0
answers
88
views
Controlling IPOPT options with pyomo doesn't work
I am using IPOPT solver for solving KKTs conditions (a bunch of equality constraints and complementarity conditions).
For assigning the solver for complementarity problem, I use the command line below:...
4
votes
0
answers
232
views
Linearize a highly non-linear objective function
[EDIT] : The formula below is updated to remove the radical, 0.5 in the term $(I_{i,v} \cdot \Delta t)$ and constant temperature $T$ replces temperature as function of current.
[EDIT] :The values of ...
- 327
-3
votes
2
answers
572
views
No value for uninitialized NumericValue object
I'm working on an optimization model in python with the pyomo library. However I'm getting an error message in python that I cannot seem to understand. The code and error message is below.
My code is
<...
4
votes
1
answer
195
views
Python library to solve nonlinear problems
What is the best python library to solve nonlinear problems? PuLP can solve only linear problems like $\max15000Z_7 + 350D_{73}Z_7 - 15000Z_8 + 350D_{86}Z_8$.
- 43
6
votes
1
answer
576
views
Is there any open source quadratic programming solver with C# API
I have a quadratic programming model (i.e., quadratic objective function and linear constraint) and, I want to solve it on an open-source solver. Since our project developed on C#, we also would like ...
- 63
4
votes
1
answer
86
views
Convex-Constrained Nonconvex-Nonconcave Minimax Problem
In the mathematical optimization theory, I have taken a glance at many papers which deal with the unconstrained convex-concave or nonconvex-concave minimax optimization, i.e.,
$$
\min_{x\in X}\ \max_{...
- 133
4
votes
2
answers
1k
views
Should all decision variables be present in Objective function?
This might be a very basic question for this community. I am reading an article and I think I have some confusion about formulating a problem. My understanding is that all decision variables should ...
- 499
2
votes
0
answers
56
views
Two-stage stochastic with non-linear recourse
I am working on a two-stage facility location problem as I described in this question.
I am solving it with the L-shaped method (Benders decomposition). The cost value between each $(i,j)$ is a ...
- 499
1
vote
2
answers
93
views
Branch and bound method for solving non-convex integer non-linear multi-objective optimization problem?
Following are the characteristics of my problem:
Objective function: two non-linear functions and one linear function
Decision variable: two integer variables ($X_1$ and $X_2$)
Constraint: three (two ...
- 159
5
votes
1
answer
282
views
What methods are used to solve multi-objective optimization problem with non-linear objective functions and integer decision variables?
Case 1: NLP
When either the objective function or at least one of the constraints or both are non-linear it is a NLP. We use generalized reduced gradient or Quadratic Programming to solve NLP. However,...
- 159
4
votes
1
answer
239
views
How to linearize a non-convex optimization objective function?
The non-convex multi-objective optimization problem in my case is defined below:
Objective 1: Minimize $f_1(X_1,X_2)=C_0+C_1(1/X_1)+C_2(X_2/X_1)+C_3X_1+C_4X_2+C_5(X_2^2/X_1)$
Objective 2: Minimize $...
- 159
4
votes
1
answer
315
views
pyomo/ipopt: nonlinear network optimization not converging
The question (very short version)
Why can I not decrease the lower boundary for the decision variable model.v_dot (see implementation) below 30 ? As soon as I do so,...
- 43
9
votes
2
answers
189
views
MINLP involving integrals, sparse matrices and CDF of random variables. Best environment?
INTRODUCTION
My research often involves solving MINLP problems with few constraints (usually two) and not many variables (say between one and three integer ones, and between one and five real-valued ...
- 191
4
votes
1
answer
97
views
Is it always possible to optimize a multivariate function sequentially?
Suppose we have a multivariate function like $f(x,y,z)$ which should be maximized with the constraints $g_i(x,y,z)\le 0 \quad \forall i$. The general rule is to use KKT conditions and derive all KKT ...
- 2,063
1
vote
3
answers
239
views
Prove NP Hardness for non-convex multi-objective optimization
The multi-objective optimization problem in my case is non-linear as it consists of three objective function of which two are nonlinear function and the third is a linear function. Lets say objective ...
- 159
1
vote
1
answer
86
views
Maximization of a differentiable and nonlinear function over a bounded space
I have a nonlinear bi-variate optimization problem like $\max \: f(x,y)$ where $f(x,y)$ is a nonlinear and differentiable function of both variables, and $0\le x\le 1$, $\:0\le y \le ub$. In order to ...
- 2,063
2
votes
1
answer
73
views
Subtracting Values from a Positive semidefinite Matrix in a Semidefinite Program
I'm trying to construct an SDP relaxation for a non-convex quadratic program ($x^{\intercal}\mathbf{H}x$) with the following objective function:
\begin{equation}
x_{11}y_{11} + x_{12}y_{12} + x_{21}y_{...
- 207
1
vote
0
answers
101
views
Solver issue? Xpress (slp) - Nonlinear - Python - Pyomo
I tried solving my model with xpress:
pip install xpress
And then in the model:
...
- 111
6
votes
0
answers
121
views
Water quality component optimization
I have an optimization problem that I'm attempting to tackle. As you can see in the image below, there's a graph network through which water flows. I've drawn out the problem in the image to explain ...
- 61
6
votes
1
answer
141
views
Two equivalent soft constraint implementations
Take the following optimization problem:
\begin{align}\min_x&\quad f(x)\\\text{s.t.}&\quad g(x)\le0\end{align}
with $f$ and $g$ nonlinear functions. Suppose I want to relax the constraint by ...
- 71
0
votes
1
answer
86
views
How to solve this clustering problem with heuristic or meta-heuristic approach?
I have clustering problem with servers and users. This is different to the one posted in https://math.stackexchange.com/questions/4088441/what-will-be-an-efficient-joint-clustering-solution-to-this-...
- 2,181
2
votes
1
answer
178
views
Which linearisation technique is correct?
I have the objective function (Maximally Diverse Grouping Problem) as
$$\max\sum_{g=1}^G\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}d_{ij}x_{ig}x_{jg}$$
Here, $d_{ij}$ are known parameters, and $x_{ig}$ and $x_{...
- 2,181
1
vote
1
answer
170
views
Non-linear optimization local or global solution
In an NLP, I have a constraint that I would like to formulate in a convex manner preferably without introducing binary variables and/or big M formulations if possible. The actual problem is non-convex ...
- 81
4
votes
0
answers
69
views
How can non-polyhedral sets be investigated?
To derive problem-specific cutting planes for some given problem (think something like TSP problem), one common way is to study small examples. To this end, one can create small instances for the ...
- 3,477
4
votes
1
answer
350
views
how to implement an optimization function with polynomial in Gurobi (Java)
I have the following problem:
I have an objective function with the optimization variable $x$, which looks simplified like this:
$ZF = (a+b)*(x+1)$
Here $a$ is simply a constant value.
However, behind ...
- 191
0
votes
1
answer
129
views
How to linearise this nonlinear constraint?
I have a constraint in the form
$\sum_{n=1}^{N}x_{m,n}\omega_{m,n}\ge (t_u-1)\beta_u, \forall u, u=1,2,\cdots, U$
where $x_{m,n}$ is binary variable
$t_u$ and $\beta_u$ are continuous optimization ...
- 2,181
2
votes
1
answer
113
views
How can I linearise this nonlinear proportional relation constraint?
My optimisation problem has a constraint in the form
\begin{equation}
\begin{array}{*{35}{l}}
\text{}\hspace{16.5mm}\text{ C4:} \hspace{2mm}\sum_{u=1}^U d_{u,1}L_{u}:\sum_{u=1}^U d_{u,2}L_{u}:\cdots:\...
- 2,181