# Questions tagged [nonlinear-programming]

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

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### 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.
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}\\ ...
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### 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 ...
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### 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 ...
625 views

### Deep Reinforcement Learning for General Purpose Optimization

Recently, I attended a very nice talk given by someone at the place I work about applying Deep Reinforcement Learning (DRL) for a design optimization problem. It was particularly interesting to me ...
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### 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 ...
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### How to linearize the product of two continuous variables?

Suppose we have two variables $x, y \in \mathbb R$. How can we linearize the product $xy$? If this cannot be done exactly, is there a way to get an approximate result?
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### 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 ...
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### 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 ...
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 ...
459 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 ...
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 ...
351 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 ...
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 ...
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 ...
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 ...
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### 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 $G(x)$ ...
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: ...
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 ...
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 ...
446 views

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 ...
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 ...
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?
141 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 \...
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### Trustful Nonlinear Programming

Is it possible for an NLP solver to claim that a knowingly feasible problem is infeasible? Shouldn't the solver be able to provide a solution (of course not necessarily the global optimum but a ...
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 ...
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 ...
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 ...
216 views

### pyomo - pass time limit to NEOS

I am sending a pretty complex Pyomo MINLP to NEOS using Couenne. I'm getting an error message that the solve time is too long (sorry, I don't have it still in my Python console). Is there a way to set ...
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 ...
351 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 ...
340 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 ...
210 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 ...
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 ...
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### 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 ...
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,...
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### Minimizing sum of functions with pairwise dependence

I have formulated a problem where I need to minimize the sum of $N$ functions, with only pairwise dependence between the functions (any single constraint involves only two functions having adjacent ...
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### Is a mathematical programming problem with no objective function an optimization problem?

I have a "mathematical programming" (MP) problem that does not have an objective function. Namely, I want to find a vector that satisfies all constraints (no optimization involved, right?). ...
160 views

What are the good text books to learn SQP? Are there any online courses that you can suggest?
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### 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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### 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 ...
94 views

### Does strong duality hold when I dualize only a subset of the constraints?

Suppose I know that for some non-convex program: \begin{align}\min_x&\quad f(x)\\\text{s.t.}&\quad g_i(x)\leq 0, i \in C\end{align} strong duality holds for this problem. Now, suppose I form ...
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 $N>M$ ...