Questions tagged [nonlinear-programming]

For questions about mathematical optimization problems involving a nonlinear objective function and/or nonlinear constraints.

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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,...
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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 ...
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  • 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 ...
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  • 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}...
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  • 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: $$1- 0.3^{x_{i,j}} $$
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  • 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 ...
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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=...
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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*} ...
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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 ...
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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 ...
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4 votes
1 answer
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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\} \\ \...
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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?
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  • 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....
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  • 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 $...
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  • 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. $...
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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-...
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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 ...
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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 ...
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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 ...
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  • 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 ...
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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 \\ \...
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  • 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\...
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  • 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{...
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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:...
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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 ...
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-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 <...
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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 $\max15000Z_7 + 350D_{73}Z_7 - 15000Z_8 + 350D_{86}Z_8$.
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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 ...
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  • 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_{...
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  • 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 ...
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  • 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 ...
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  • 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 ...
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  • 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,...
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  • 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 $...
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  • 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,...
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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 ...
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  • 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 ...
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  • 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 ...
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  • 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 ...
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  • 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_{...
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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: ...
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  • 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 ...
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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 ...
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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-...
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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_{...
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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 ...
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  • 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 ...
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  • 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 ...
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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 ...
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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:\...
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