Questions tagged [nonlinear-programming]
For questions about mathematical optimization problems involving a nonlinear objective function and/or nonlinear constraints.
212
questions
1
vote
1
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64
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Maximizing sum of probabilities with variable distributions
Suppose $\\{X_i\\}$ are binary decision variables and $\\{A_j\\}$ are Skellam random variables with $(\mu_1, \mu_2) = (\sum_i b_{i} X_i, c_j)$. Here, $b_i, c_j \in \mathbb{R}^{\geq 0}$ are constants. ...
- 111
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0
answers
53
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Multilinear programming over the simplex
Let $\triangle_3 \in \mathbb{R}^3$ be the $3$-simplex. I am solving a series of multilinear programming problems that looks like this:
$$\text{Maximize}\sum_{0\leq i, j, k \leq 3} A_{i,j,k} x_i x_j ...
- 151
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votes
0
answers
463
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How to write this objective in CVXPY for quasiconvex programming?
I have the following objective that I want to maximize:
\begin{equation}
\max_{U_T\in \mathbb{R}, x\in\mathbb{R}^T} J_\alpha(U_T) = \frac{\alpha}{\alpha-1}\log\left(\frac{\cosh(U_T)}{\cosh(\alpha U_T)^...
- 86
2
votes
0
answers
94
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log-log regression as reward function in optimization problem
Consider the model $\hat{y}_t = e^{\text{trend} + \text{seasonality}} \prod_k^K x_{k, t}^{b_k}$
where $K$ denotes different investment alternatives. You can think that trend and seasonality are ...
- 21
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0
answers
28
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Steepest ascent vector at a point of a constrained nonlinear problem
I'm looking at this article:
"Packing unequal circles into a strip of minimal length with a jump algorithm" (Stoyan et Yaskov, 2014) DOI
In section 5, a nonlinear constrained model is ...
- 2,495
2
votes
1
answer
129
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How to model a penalty for exceeding a threshold in a nonlinear optimization problem using IPOPT?
I'm working on a nonlinear optimization problem where I have a decision variable representing my product's price (P_m) and a constant representing my competitor's price (P_c). I want to introduce a ...
- 163
3
votes
1
answer
60
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Finding a starting ellipsoid and a minimum volume to approximate a convex optimization problem
Suppose we have a convex optimizatiom program:
\begin{align}
\min &\quad f_0(x)\\
s.t. &\quad h_i(x) = 0 && i=1,\ldots, p\\
&\quad g_i(x) \leq 0 && i=1,\ldots, m\\
&\...
- 53
0
votes
1
answer
89
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Algorithms for maximizing the sum of power functions with linear constraints?
I’m working on an optimization problem that arises from maximizing the return obtained from investing in different marketing levers. The return from each lever exhibits diminishing returns, and is ...
1
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1
answer
66
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Knitro dimension of lambda for Hessian
I'm trying to supply knitro with a Hessian but struggle to understand the dimension of the Lagrangian multiplier $\lambda$. From my general education and knitro's ...
- 13
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0
answers
15
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Problem in understanding an equation from a paper about iterative Linear-Quadratic Regulator
I'm reading a paper about iterative Linear-Quadratic Regulator (iLQR) and there are a lot of points that I don't understand.
https://homes.cs.washington.edu/~todorov/papers/TassaICRA14.pdf
I think ...
- 101
2
votes
1
answer
70
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Question About Fritz John Theorem and Slater Constraint Qualification
Background Information
I am studying constraint qualifications. Here are two theorems leading to my question:
Theorem 1$\space\space\space\space$ [Fritz John Theorem] Suppose that $f, g_1, \dots, g_k$...
- 123
1
vote
1
answer
50
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Is there any literature on constrained nonlinear optimization where constraint returns have to be queried from an oracle?
I am looking at literature considering constrained optimization problems of the form:
$\min_{x\in X\subseteq R^n} f(x), \text{ subject to } g_{oracle}(x) \leq 0$
The optimization algorithm doesn't ...
- 13
-1
votes
2
answers
181
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How do I optimize this problem where the constraints and objective are variable?
Problem Definition:
Pa = Constant
Pb = Constant
Vmax_a = Constant
Vmax_b = constant
Objective Function:
...
-1
votes
2
answers
105
views
How to maximize sum of cosine squared plus sum of sine squarred?
I want to maximize this function
$$\left(\sum_{k=1}^{N}\cos(2\pi f_1t_k+\phi_k+\alpha\pi(k-1))\right)^2+\left(\sum_{k=1}^{N}\sin(2\pi f_1t_k+\phi_k+\alpha\pi(k-1))\right)^2,$$ where the variables are $...
- 381
2
votes
1
answer
66
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Derivative-free based optimization subject to a linear constraint
I'm dealing with a NLP-problem that can be formulated as:
$$\min_{\overrightarrow{x}} f(A\cdot\overrightarrow{x})$$, where $\overrightarrow{x}$ is a vector of $n$ design points where every element ...
- 1,043
0
votes
1
answer
51
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Knitro`ms_maxsolves` equivalent in Ipopt
Ipopt/Knitro are local optimization solvers, so for nonconvex problems convergence doesn't guarantee optimality. However, Knitro has a multi-start method where one can start with more random initial ...
- 3,980
0
votes
1
answer
59
views
Min-convex function as constraint
I have a constraint that is as follows:
$$
Ax - f(x) \leq 0
$$
where $f(x)=min_y(g(x,y))$. Which is convex. I can even get the gradient in $x$. How can I reformulate my constraint? or what ...
- 507
1
vote
1
answer
45
views
$\min\{f(x_1),\dots,f(x_n)\}$ with other constraints
I have an optimization problem which goes:
\begin{align*}
\text{Minimize:}
\\
& \sqrt{x} + \sqrt{y} \tag{NL-objective}
\\
\text{Subject to:}
\\
&3x + 2y \geq 2 &...
- 11
0
votes
3
answers
168
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Using PULP to model machines in a factory
Complete revision:
I have all 4 machines that can run in positive and negative directions which results in 2 outputs:
$ P_1 = \begin{cases}
-390 \le P_1 \le -300 & \text{for neg mode}\\
0 &...
- 1
2
votes
2
answers
123
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Optimize least squares penalized by curvature of log pdf
I have probability values $p \in \mathbb{R}^n$. Given $A \in \mathbb{R}^{m\times n}$, $b \in \mathbb{R}^m$, I want to minimize the following objective function. $||Ap - b||_2^2 + \sum_{i=1}^{n-2} (\...
- 121
2
votes
2
answers
97
views
Potential methods for solving quadratic optmization problem
I am trying to solve a non-convex optimization problem with the help of sequential quadratic programming.
I need to develop an algorithm inside SQP to solve this subproblem. What potential methods (...
- 67
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votes
0
answers
56
views
How do I solve this non-linear optimisation problem based on simulations?
I have an optimisation problem that is essentially a knapsack problem with a non-linear objective.
I have an input dataframe that contains a row for each item, each item has columns defining its mean ...
- 31
1
vote
1
answer
93
views
Sensitivity analysis for decision vectors in convex programming
Can we perform sensitivity analysis on the decision variables for the perturbed right-hand side of the constraints in a convex/nonlinear program? I know a basic result regarding the sensitivity of the ...
- 77
0
votes
0
answers
68
views
Resource selection problem with non-linear objective function
I have an optimisation problem to solve but I can't model it correctly. Any insight is welcome :)
It has been a few years since my optimisation classes in university, and while I have forgotten a lot ...
- 1
3
votes
0
answers
77
views
Looking for an efficient way to solve a fractional problem (affine function over euclidean norm )
While working on optimization issues I encountered the following problem:
$$\left\{\begin{array}{ll}
{\displaystyle \sup_{z\in\mathbb{R}^{m}}} &\frac{ \langle c,z \rangle + \rho}{ \left\|B z\...
- 151
2
votes
3
answers
169
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Linearization the product of three variables (two binary & one continuous)
Consider the following binary variable $x \in \{0,1\}$ and two continuous real variables $y,p \in \mathbb{R}$.
I am trying to model the following conditional equations as constraints:
\begin{cases}
...
- 103
1
vote
4
answers
141
views
How to perform clustering of a large number of nodes?
I have a clustering problem with around 400-500 nodes. The edge between any two nodes has a weight (between 0 and 1, 0: can be considered as there is no edge/connection between these two nodes) as ...
- 2,211
1
vote
2
answers
109
views
My barrier function is always giving a complex number
I am working on implementing the interior point method, and the barrier function always gives me a complex number. B(x) = f(x) - t * sum(ln(hi(x))). I have changed the value of 't' to see the B(x) ...
- 67
1
vote
2
answers
574
views
Non linear programming
I want to solve a large scale non linear optimization problem and there are two methods interior point method and sequential quadric programing usually used to solve non linear optimization problem. I ...
- 67
2
votes
2
answers
337
views
I want to solve an optimization problem with nonlinear piecewise objective function (I tried Pyomo with "ipopt" solver but I had an error)
I want to solve an optimization problem where the objective function is the summation of nonlinear, piecewise functions in the decision variables q_i's such that when a decision variable q_i < 1, ...
- 21
1
vote
1
answer
163
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Nonlinear fractional objective function
Could you please teach me when an optimization model with fractional terms in the objective function can be linearized or solved optimally?
I only know that if the objective function has a single ...
1
vote
2
answers
61
views
Solving single-variable fractional equation
Can anyone advise how this nonlinear equation with a single variable $x$ can be solved as a closed form?
${\left(\frac{x}{1-x}\right)}^2.{\left(\frac{x-C}{\left(1-x\right)-N}\right)}^2=H.\frac{2x-C}{2\...
2
votes
1
answer
122
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Formulation of nonlinear nonconvex optimization problem and finding appropiate solver
Consider the notation and objective below for this sequential resource allocation problem:
Allocation channels $i \in (1, 2)$
Spend/Cost timestep i channel j: $C_{i, j}$
Total resource: $B$
Horizon: $...
3
votes
1
answer
152
views
Modelling a decision variable as an index of a (fixed) set
I'm trying to model the following MINLP problem in Pyomo.
We are trying to minimize a nonlinear objective function $f$ in $x_i \in \lbrace{0, 1, 2\rbrace}$ for $i= 1, 2, \dots, N$ and subject to a ...
- 1,043
2
votes
1
answer
239
views
Implementing a MINLP problem in Pyomo: giving an infeasible solution
I'm trying to implement a MINLP problem which is described in a previous post here: How do we formulate a problem where the decision variable has an index that is also a decision variable?
The only ...
- 1,043
6
votes
3
answers
463
views
How do we formulate a problem where the decision variable has an index that is also a decision variable?
I want to maximize the sum of a nonlinear function $f(.)$ w.r.t. $x$ that is convex in $x$:
$$\max \sum_{i=1}^N f(x_i), $$where $x_i$ is a continuous variable and $0 \le x_i < 1$ for $i = 1, 2, \...
- 1,043
1
vote
1
answer
534
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OR-Tools: Nonlinear constraints?
I have inherited a reasonably simple ortools-based optimizer (Python) with linear relationships that I need to non-linear-ize, and I have no idea how to do that.
The relevant part of my problem looks ...
- 119
1
vote
1
answer
131
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Formulating a continuous NLP problem with a class variable
In this minimization problem we have $N$ items, $j= 1, 2, \dots, N$ and a decision variable $x_j$ which are continuous values.
For every item, we have a nonlinear objective function $f$ in function of ...
- 1,043
5
votes
1
answer
261
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Optimize selection of metal sheets to keep in stock
I already asked this on stack overflow but just found this forum instead and figured it was more suited here. If this isn't allowed please feel free to tell me and I'll delete the post.
I am doing ...
- 53
-1
votes
2
answers
89
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How to apply smooth approximation to non-linear complementarity constraints?
$P =$
$ x, if U \geq U^{max} $
$ y, if U^{up} < U < U^{max} $
$ z, if U^{down} < U < U^{up} $
$ \alpha, if U^{min} < U < U^{down} $
$ \beta, if U \leq U^{min} $
Where $P$, and $U$ ...
2
votes
0
answers
72
views
Branching the product of binary and continuous variable in Gurobi
I have a binary variable (X) multiplying a continuous variable (Y). I know I can linearize by adding an auxiliary variable (I have that model working), but I now want to do my own branching in the ...
- 507
2
votes
1
answer
94
views
Formulate revenue maximization problem and find an appropriate solver
I am trying to maximize expected revenue over a horizon.
Consider the following function:
\begin{align}
sales(budget_1, budget_2) = \sum_te^{C_1t} * budget_1t^{saturation_1t} + e^{C_2t} * budget_2t^{...
4
votes
1
answer
174
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How to solve a "nearly" linear program
Given a positive integer $n$, a constant $k=2/3$, and $7$ variables $x_1, x_2, x_3, x_{12}, x_{13}, x_{23}, x_{123}$ (non-negative reals or integers) I would like to find:
$$\min \binom{x_1}2$$
...
- 393
2
votes
1
answer
112
views
How to mathematically model this multi-objective optimization problem?
I have a system of $M$ machines and $U$ users.
Each machine has a capacity in terms of number of resources.
Let, machine $m$ has $\zeta_m$ resources.
Each user has a service demand $d_u$ and there is ...
- 2,211
3
votes
1
answer
175
views
How do I pass an objective bound to Gurobi?
I have a non-convex Quadratic Programming over unite simplex set. I have a valid lower bound on the objective function (goal is minimization problem).
If I add a constraint like
$$f(x)\geq lower~bound,...
1
vote
0
answers
56
views
How to avoid complementarity constraints in continuous nonlinear program?
In my two-stage continuous NLP problem, I have a constraint in second stage:
$X_{g,k}$ = $X_{g,0} + a_{g} d_{g} $, if $X_{g,k} \in [X_g^u,X_g^l]$
$X_{g,k} = X_g^u$, if $X_{g,k} \geq X_g^u$
$X_{g,k} ...
1
vote
1
answer
65
views
An if-then-else logic whose condition is an inequality
I was hoping to get some help in modelling the following logic. I tried to solve it by "Big-M method" but failed. Thank you in advance!
$a(k,n)$ and $b(k,n)$ are known constants, $\lambda$ ...
- 33
-1
votes
1
answer
122
views
Ipopt finds a better solution if I do not eliminate the zeros in the hessian matrix ?(we eliminate the zeros by defining the structure)
I use Ipopt to solve a problem with sparse hessian and jacobian matrices.
If I provide the hessian matrix: its structure, and the non zeros elements in the hessian matrix, it will be really fast.
If I ...
- 391
0
votes
1
answer
106
views
The max_wall_time and max_cpu_time in ipopt are not working?
The max_wall_time and max_cpu_time are not working in ipopt (cyipopt).
See example:
...
- 391
0
votes
2
answers
98
views
Bellman Equation for nonlinear model
Consider the following model:
\begin{align*}
max \quad Z &= 19x_1 - 3x_1^2 + 5x_2^2 - x_2^4 + 4x_3 \\
& s.t. \quad x_1 + 3x_2 + 3x_3 \leq 7 \\
& \quad \quad \quad x_1,x_2,x_3 \geq 0
\end{...
