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Questions tagged [nonlinear-programming]

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

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1 vote
1 answer
34 views

Modeling $-\ln(1 - w \cdot \sigma(x))$ as disciplined convex programming

Given $0 < w \leq 1$, I would like to use the function: $$ -\ln(1 - w \sigma(t)), $$ where $\sigma(t) = 1 / (1 + \exp(-t))$ is the sigmoid function, in my objective. It's a bit tedious, but this ...
2 votes
1 answer
95 views

Are there classes of non-linear programs that always have sparse solutions?

It is well-known that linear programs always have sparse solutions -- solutions in which only at most $m$ variables are nonzero, where $m$ is the number of constraints. In the answers to this question,...
2 votes
2 answers
174 views

Are McCormick Envelopes exact for the following class of optimization problems?

I have the following optimization problem: \begin{align*} \text{minimize} \quad &\mathbf{c^T x} \\ \text{such that} \quad &\mathbf{x} \in S. \end{align*} Here, $S$ is a polyhedron of the form $...
1 vote
2 answers
172 views

Linear approximation of fraction for a maximization problem

I have a problem given as \begin{align*} \underset{\mathbf{x}}{max} & \left|\mathop{\sum_{n=1}^{N}}\left[\frac{\mathbf{a}\left(n\right)}{\mathbf{b}\left(n\right)+\mathbf{x}\left(n\right)}\right]\...
1 vote
1 answer
123 views

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. ...
0 votes
0 answers
51 views

Modeling Approach to Adjust linear Elasticity Effect in Pricing Optimization

I am working on a pricing optimization model for a product where the price depends on the competition as well as our costs. The current formulation of the model is: ...
1 vote
1 answer
45 views

Converting a function composing of multipe pieces into a linear equation

I have a variable (alpha) which depends on some other binary variables, denoted as X_i. So, for some combination of other variables, alpha may take a value (Beta_j). I added some auxillary variables (...
1 vote
1 answer
795 views

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 ...
1 vote
1 answer
212 views

How do I solve this Optimization problem?

Optimization of a simple expansion problem minimise: $$ \sum_{t=1}^{5}\left[\sum_{i=1}^{2}x_{i,t}CC_i\left(\frac{1+EIC}{1+r}\right)^t+UE_t*C_{UE}\right] $$ subject to: $$ 0 \leq x_{i,t} \leq 5 \\ ...
1 vote
0 answers
87 views

volume-weighted mean equality constraints

I have the following optimization objective function for a dynamic pricing problem: \begin{align*} sum\_profit = \sum_{i \in sales\_point} \Bigg( constant[i] + {elasticity[i]} \cdot (movement[i] + ...
0 votes
0 answers
49 views

Solving convex separable programming problem using interior point method?

In my engineering application, all decision variables are non-negative and everything is convex separable. In addition to that, the only function that I am trying to approximate with grid point are $f(...
2 votes
1 answer
135 views

Lifting a 3rd order polynomial into a higher dimensional space

An MINLP from a paper I am reading has the following expression in its constraints: $$ p_{l,s}=z_lb_l\Delta\theta_{l,s}+b_l\lambda_{l,s}u_l\Delta\theta_{l,s} $$ Where from left to right: $p_{l,s}$: ...
2 votes
1 answer
213 views

How to transform a binary QP into an MILP?

I have a binary quadratic problem with objective ${\bf{x}}^T{\bf{Qx}}+{\bf{c}}^T{\bf{x}}$ subject to ${\bf{A}}{\bf{x}}\le{\bf{b}}$ ${\bf{A}}_{eq}{\bf{x}}={\bf{b}}_{eq}$. here ${\bf{x}}$ is binary. ...
3 votes
2 answers
144 views

Lagrangian Multipliers for constraints in nonlinear optimization problems?

Suppose I want to optimize some function of continuous variables and the objective is nonlinear; in this context, gradient-based methods are quite popular. To my knowledge, soft constraints can be ...
-1 votes
2 answers
172 views

How to describe nonlinear programming in gurobipy?

The optimization task The optimization task concerned here is: A square matrix $A\in \mathbb{R^{5\times 5}}$ satisfies the condition: the elements of the first row all are 1, the elements of the ...
1 vote
2 answers
75 views

How to maximise black-box function defined on a subset of integers, with access to its derivative?

The basic form of my problem is as follows. Let $S = \{ 1, 2, ..., 30 \}$. I have an integer-valued function $f$, whose argument is an integer-valued vector $\vec{x} \in S^{200}$. I also know that ...
5 votes
2 answers
577 views

Transform nonlinear cost function to get LP or MILP

I'm trying to schedule power of multiple prosumers in a microgrid. The problem includes a cost function with min and max ...
1 vote
0 answers
101 views

Converting a Linear Program with TU Constraint Matrix to a Nonlinear Convex Model: Solver Performance?

I'm currently working on a large Mixed Integer Program (MIP) where the constraint matrix is Totally Unimodular (TU), allowing me to model it as a Linear Program (LP) for efficiency, as total ...
1 vote
0 answers
94 views

Numerical infeasibility for moving numbers along some specific conversion edges to get maximal number on target node from some starter node

I have a problem that can be represented as an optimization problem. Sometimes, solver engines report infeasible depending on the parameters I have at hand. The root cause is numeric ranges. ...
0 votes
0 answers
43 views

Using Knitro or Xpress SLP via FICO Xpress Python API for Local and Global Solve Methods

Can someone guide me on how to utilize the FICO Xpress Python API to invoke Knitro or Xpress SLP, specifically for choosing between local and global optimization methods? I am referring to the version ...
3 votes
1 answer
317 views

How to convert non-normal probabilistic constraints to deterministic ones for mathematical modelling?

I am working on a chance-constrained optimisation model that takes into account uncertainty. I am aware of how to convert constraints that are of a probabilistic nature into the equivalent ...
0 votes
0 answers
58 views

Better formulation of bilinear terms

I am working on an optimization problem where I need to formulate a constraint that represents the total sales value under specific conditions. The challenge lies in creating an expression that ...
1 vote
2 answers
219 views

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 votes
2 answers
138 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 $...
18 votes
8 answers
2k views

Are metaheuristics ever practical for continuous optimization?

All of the applications of metaheuristics that I can think of are for discrete optimization (usually combinatorial optimization) problems. Are metaheuristics ever practical tools for continuous ...
5 votes
0 answers
550 views

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)^...
2 votes
0 answers
98 views

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 ...
0 votes
0 answers
63 views

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 ...
0 votes
1 answer
93 views

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 ...
2 votes
1 answer
79 views

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 $f, g_1, \dots, g_k$...
1 vote
0 answers
28 views

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 votes
1 answer
171 views

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 ...
3 votes
1 answer
67 views

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\\ &\...
1 vote
1 answer
81 views

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 ...
0 votes
0 answers
17 views

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 ...
1 vote
1 answer
62 views

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 ...
-1 votes
2 answers
186 views

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: ...
2 votes
1 answer
70 views

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 ...
0 votes
1 answer
68 views

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 ...
0 votes
1 answer
74 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 ...
1 vote
1 answer
57 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 &...
4 votes
1 answer
501 views

Free solver for MINP problems

I have a mixed-integer nonlinear programming (MINP) problem. Is there a free solver for such a problem?
0 votes
3 answers
216 views

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 &...
2 votes
2 answers
130 views

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} (\...
2 votes
2 answers
110 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 (...
0 votes
0 answers
62 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 ...
1 vote
1 answer
126 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 ...
0 votes
0 answers
72 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 ...
14 votes
4 answers
2k views

CPLEX non-convex Quadratic Programming algorithms

CPLEX solves non-convex quadratic problems to global optimality with a global optimality option (in version 12). The relevant pages are this and this. I benchmarked many solvers, and see that CPLEX ...
3 votes
1 answer
457 views

About Function Manipulation

I have a function as follows (updated after the clarification question): $$\max_{x∈X}\left(\sum\sum c_{ij}x_{ij}-\max_{y∈Y}\sum\sum d_{ij}x_{ij}y_{ij}\right)$$ where $x_{ij},y_{ij}$ are decision ...

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