Questions tagged [nonlinear-programming]

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

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4
votes
2answers
153 views

Is There Another Way To Code The Idea of a MAX Constraint Without The Use of Binary Variables?

I have a constraint of the following form that describes the growth of trees, where the population of trees in period $t$ is the previous period's population minus some trees infected with a virus: $...
4
votes
0answers
55 views

Identifying saddle point in constrained optimization

Suppose we are minimizing $f(x)$. The first order necessary condition of $x^*$ being local minmum is: $$\nabla f(x^*)= \mathbf{0}.$$ For sufficiency, we check if also $\nabla^2f(x^*) \succ 0$, i.e., ...
6
votes
3answers
1k views

Is it abnormal for a model to take 8+ hours to solve?

I am building my first optimization model, it is quite large and also a non-linear problem. I have had my model solving on the NEOS Optimization Server and after 8 hours of trying to solve, the server ...
3
votes
1answer
121 views

Rolling Horizon Methods in Knitro

I am trying to solve an NLP using the Knitro solver, but I am beginning to think that I will not be able to solve the model because it is too complex. I have heard that "rolling horizons" can be used ...
5
votes
2answers
125 views

Minimizing $x_1/x_2$ over a simplex in the positive orthant

I need to solve the following problem \begin{align}\min&\quad x_1/x_2\\\text{s.t.}&\quad Ax \leq b\\&\quad x > 0\end{align} where $A$ is a positive matrix. The best thing I can think ...
4
votes
1answer
116 views

Software for multi-objective optimization

I am looking to solve a multi-objective chance-constrained blending problem. Are there any suggestions about the software to use to try and solve a problem like this?
3
votes
1answer
92 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 ...
6
votes
1answer
78 views

Nonsmooth constrained convex optimisation: convergence results?

I am working on a projection problem on a very large set of highly related constraints: \begin{align} \min_x & \quad\|x-x_k\|_2^2 \\ \mathrm{s.t.} & \quad\max_{T\in\mathcal{T}} \sum_i \frac{...
12
votes
1answer
99 views

Benders subproblem feasible region dependent upon solution master problem

Suppose I want to solve a naturally MINLP problem of the following form: $$ \min_{x,y} \{c'x + y \mid Ax \leq b, Dx + Ey \leq f, G(x)y\leq g, x \in \mathbb{Z}, y \in \mathbb{R}^+\} $$ Here $G(x)$ ...
8
votes
1answer
209 views

Can Gurobi or CPLEX handle nonlinearly constrained problems?

Though my title is quite general (please feel free to edit), indeed, I wonder if the following models can be solved in Gurobi or CPLEX. Model 2 is just an alternative one to Model 1. Although we ...
4
votes
1answer
81 views

Dealing with a non-convex problem

I have the following objective function. The variables: $h_p$, $e_{trs}\left(h_p\right), w_{trs}\left(h_p\right)$ are all non-negative continuous. $T,R,S,\pi_{trs}$ are polynomially-sized sets. All ...
6
votes
4answers
160 views

Sequential quadratic programming source

What are the good text books to learn SQP? Are there any online courses that you can suggest?
5
votes
1answer
101 views

How to minimize a weighted sum of RMSE-like terms?

I am trying to solve the following problem: \begin{align} \min&\quad f(x) = \sum_{i=1}^{n}{a_ix_i} + \sum_{i=1}^{n}{b_i\sqrt{\sum_{j=1}^{m}{\left(y_{i,j}-x_i\right)^2}}}\\\text{s.t.}&\quad x_{...
8
votes
2answers
350 views

How can I transform this MILP into an LP problem?

I have a MILP problem with one of the constraints is given below. Sometimes, even for a small-sized problem, the solver takes a very long time to find a solution. What could be an efficient ...
12
votes
1answer
295 views

Linearization of the product of two real valued variables - Binary expansion approach

I want to minimize the following objective function: \begin{align}\min &\quad x\cdot y\\\text{s.t.}&\quad2 \le x \le 5\\&\quad5 \le y \le 10.\end{align} Since the objective function is ...
12
votes
2answers
236 views

Can Tuning Knitro Solver Considerably Make A Difference?

I have an NLP that I am hoping to solve with Knitro and I am aware of a multitude of different settings that you can "tune" in order to improve solution performance. I am not familiar with ...
11
votes
1answer
372 views

Suggested Resources for Non-Linear Optimization

I recently completed an undergraduate course in Linear Programming and Operations Research. I am willing to look into advanced concepts and Non-Linear Optimization algorithms and also, their method of ...
7
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1answer
165 views

Is this formulation linear or non-linear?

Can you help me figure out if this formulation constitutes a non-linear problem? I believe It is a linear problem but my solver (GAMS) is unable to produce a acceptable solution. $x,y$ and $\text{...
7
votes
2answers
448 views

Why does a Max constraint work, but this non-negativity constraint does not?

Suppose I have the following constraint: \begin{align}x_{t} &= x_{t-1} + y_{t-1} - z_{t-1}\\x_{t} &\ge 0\end{align} From my limited experience in coding my own problem, I have found that my ...
9
votes
1answer
168 views

Solving convex programs defined by separation oracles?

General question: What software can solve convex programs defined by a separation oracle? The objective function is concave, and the feasible set is a polytope. By a separation oracle I mean that I ...
5
votes
0answers
33 views

In a binary logistic regression context, how to introduce a constraint to model the dependency between consecutive samples

Imagine we are running a logistic regression to identify opportunities for car sale promotion, using previous promotion campaign's result. Each $y$ is the increase of car sale after the promotion. ...
11
votes
1answer
446 views

Solvers and saddle points

It seems like most solvers that can tackle nonlinear nonconvex optimization problems (e.g. IPOPT) operate on ultimately solving for the first-order optimality conditions. Can it therefore be assumed ...
10
votes
3answers
410 views

Parallel nonlinear solvers

I've noticed that parallel (CPU or GPU) nonlinear programming solvers are few and far between. It seems that if any parallelization is involved at all, it generally applies to solving the underlying ...
8
votes
1answer
339 views

Do the KKT conditions hold for mixed integer nonlinear problems?

I was wondering if the KKT conditions are applicable for for MINLPs, and if not, why not? What about the case when the integer variables are modeled using constraints involving just continuous ...
11
votes
2answers
478 views

Dedicated solver for convex problems

Are you aware of a fast solver (open source or commercial) for convex NLPs that is faster than IPOPT? I'm interested in problems in the 50K+ variable range, both dense and sparse. Ideally, it would be ...
16
votes
1answer
911 views

IPOPT with HSL vs MUMPS

What are the advantages (if any) of using IPOPT with HSL vs MUMPS? HSL has a reputation of being faster, but does it walk the walk? In particular, does HSL scale better for large-scale problems? We ...
11
votes
0answers
140 views

Characterizing the solution of a (non) linear maximization program

I have the following maximization program \begin{align} \max\limits_{\{q_i\}}&\quad\sum\limits_{i=1}^nq_i \\ \text{s.t.}&\quad\begin{cases} k_j \geq \sum\limits_{i=1}^n q_i^{1 \over \...
7
votes
1answer
113 views

Solutions to a parametrized optimization problem

I have the following maximization program \begin{align} \max\limits_{\{q_i\}}&\quad\sum\limits_{i=1}^nq_i \\ \text{s.t.}&\quad\begin{cases} k_j \geq \sum\limits_{i=1}^n q_i^{1 \over \...
12
votes
2answers
174 views

Linearisation techniques for MINLPs

I am wondering what kinds of linearisations people do for MINLPs outside my field of expertise. I work in global optimisation, so by "linearisation" we would typically mean one of the following: ...
9
votes
3answers
1k views

Matlab fmincon for a problem with many nonlinear constraints

Using Matlab to solve a problem which has linear objective function and many nonlinear constraints, I am trying to generate the inequality nonlinear constraints by a function and pass it to fmincon ...
13
votes
1answer
459 views

How to formulate a problem to prove/disprove convexity?

Given a general non-linear problem: \begin{align}P:\qquad&\min_{x\in X} f(x)\\\text{s.t.}\qquad&g(x)\leq 0\end{align} where $f$ is a non-linear function, $g$ is a vector of non-linear ...
4
votes
0answers
193 views

How can I formulate this multi-objective optimization problem?

Now, for each system $X$ $(X=A,B,C,E)$, my objective is $$\max\min\frac{s_{x_u}}{d_{x_u}}$$ here, $x=a$ for system A, $x=b$ for system B and follows... and for the whole system, my objective is $$\max\...
12
votes
2answers
350 views

Expressing an implication as ILP where each implication term comprises a chain of boolean ORs

Consider an implication of the form $A \implies B$ where both $A, B$ comprises a chain of Boolean OR variables. For example, $(a_1 \lor a_2 \lor a_3) \implies (b_1 \lor b_2 \lor b_3)$. How can this ...
6
votes
1answer
136 views

Obtaining the intermediate solutions in AMPL

I know that for some solvers, for example, the constraint programming solver in Google OR-Tools, it is possible to see all the intermediate solutions that the solver finds while it searches for an ...
22
votes
4answers
2k views

Linearize or approximate a square root constraint

I encounter a nonlinear constraint that contains the square root of a sum of integer variables. Of course one could use nonlinear solvers and techniques; but I like linear programming. Are there any ...
14
votes
4answers
3k views

NLP solvers in pyomo other than ipopt?

I am solving a highly constrained (large number of constraints and large number of variables, but small degree of freedom) NLP problem, and for start, I was using Matlab's ...
8
votes
1answer
265 views

Disciplined convex programming representation of $x\cdot\min x$

How can I reformat the problem below to follow DCP rules? DCP rules are Disciplined Convex Programming Rules that allow convex programs to be solved. DCP Is there a way to reformat the problem ...
6
votes
1answer
285 views

How to reformulate (linearize/convexify) a budgeted assignment problem?

I have a scheduling problem at hand. In my system, there is a service station with $M$ service outlets, therefore, the service station can serve $M$ users at a time. But, there are $N$ users $N>M$ ...
24
votes
5answers
4k views

Cubic programming and beyond?

It is almost inevitable in Operations Research to come across linear or quadratic programming problems. The overall structures of these problems are below: \begin{align}\begin{array}{ll} \sf{Linear}\\ ...
10
votes
1answer
419 views

Termination Criteria of Solver in Pyomo

I am solving a nonlinear optimization problem using Pyomo with Ipopt as solver. The solver exits with the status: EXIT: Optimal Solution Found. This I can cross ...
13
votes
1answer
457 views

Sum of Max terms maximization

Maximizing sum-of-max terms is an NP-hard problem. The objective function is a convex function and maximizing a convex function is a hard problem. Also, this is a non-differentiable function. CPLEX ...
11
votes
1answer
154 views

Heuristics for mixed integer linear and nonlinear programs

What are some primal heuristics that mixed-integer linear and nonlinear program solvers use to quickly obtain a reasonably good feasible solution?
6
votes
1answer
73 views

Minimizing a project costs through nonlinear optimization

I have a project and I want to minimize the costs. I am are responsible for the inspection of 1000 miles of sewer grid in Canada. My goal is to provide time high quality inspection reports. I tried to ...
7
votes
1answer
369 views

KKT inequality conditions

Let's say I have an objective function $$f(x_1,x_2, \cdots, x_n)$$ and $N$ constraints $$x_i \ge 0. $$ I am trying to solve it with KKT conditions. Now the objective function becomes $$f(x_1,x_2,...
14
votes
1answer
2k views

How to formulate (linearize) a maximum function in a constraint?

How to formulate (linearize) a maximum function in a constraint? Suppose $C = \max \{c_1, c_2\}$, where both $c_1$ and $c_2$ are variables. If the objective function is minimizing $C$, then it can be ...
18
votes
8answers
1k 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 ...
12
votes
1answer
348 views

When is the original BFGS algorithm still better than the Limited-Memory version?

I have been going through Andrew NG's original data science course on Coursera. I learned the BFGS algorithm at some point in my OR education, but not the Limited Memory version that Andrew NG focuses ...
14
votes
4answers
911 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 ...
12
votes
1answer
289 views

McCormick envelopes and nonlinear constraints

I have a problem with a nonlinear constraint. The non-linearity stems from a term of the form $xb$, where $x \in \mathbb{R}^+$, $x < M$ and $b \in \{0, 1\}$. I am able to remove this non-linearity ...
40
votes
8answers
1k views

Optimization Problem Libraries

Can someone please make a list of optimization problem libraries so that the community can add to and refine it? I know a few off the top of my head.