When is a problem infeasible

A zero preference reflects that the constraint or bound cannot be relaxed. It is the responsibility of the modeler to provide preferences that yield a feasible relaxed problem.

Note, that if all preferences are nonzero, the relaxed problem is always feasible with the exception of problems containing binary or semi—continuous variables, since because of their special associated modeling properties, such variables are not relaxed.

Note, that this utility does not repair the infeasibility of the original model, but based on the preferences provided by the user, it introduces extra freedom into it to make it feasible, and minimizes the utilization of the added freedom.

The magnitude of the preferences does not affect the quality of the resulting solution, and only the ratios of the individual preferences determine the resulting solution. Using the new variables introduced, it is possible to warm start the primal simplex algorithm with a basic solution. An infeasible, but first phase optimal primal solution typically speeds up the solution of the relaxed problem. Once the optimal solution to the relaxed problem is identified and is automatically projected back to the original problem space , it may be used by the modeler to modify the problem in order to become feasible.

However, it may be of interest to know what the optimal objective value will be if the original problem is relaxed according to the solution found be the infeasibility repair function. In order to provide such information, the infeasibility repair tool may carry out a second phase, in which the weighted violation of the constraints and bounds are restricted to be no greater than the optimum of the first phase in the infeasibility repair function, and the original objective function is minimized or maximized.

While such a relaxation allows considering the effect of the original objective function in more detail, on some problems the trade—off between increasing delta to improve the objective can be very large, and the modeler is advised to carefully analyze the effect of the extra violations of the constraints and bounds to the underlying model. Note, that it is possible that an infeasible problem becomes unbounded in the second phase of the infeasibility repair function.

In such cases, the cause of the problem being unbounded is likely to be independent from the cause of its infeasibility. When not all constraints and bounds are relaxed it is possible for the relaxed problem to remain infeasible. In such cases it is possible to run the IIS tool on the relaxed problem, which can be used to identify why it is still infeasible.

It is also possible to limit the amount of relaxation allowed on a per constraint side or bound by using XPRSrepairweightedinfeasbounds. It can sometimes be desired to achieve an even distribution of relaxation values. This can be achieved by using quadratic penalties on the added relaxation variables, and is indicated to the optimizer by specifying a negative preference value for the constraint or bound on which a quadratic penalty should be added.

In rare cases a MIP problem can be found to be infeasible although its LP relaxation was found to be feasible. In such circumstances the feasible region for the LP relaxation, while nontrivial, contains no solutions which satisfy the various integrality constraints. These are perhaps the worst kind of infeasibilities as it can be hard to determine the cause. In such cases it is recommended that the user try to introduce some flexibility into the problem by adding slack variables to all of the constraints each with some moderate penalty cost.

A problem is said to be unbounded if the objective function may be improved indefinitely without violating the constraints and bounds. This can happen if a problem is being solved with the wrong optimization sense, e. However, when a problem is unbounded and the problem is being solved with the correct optimization sense then this indicates a problem in the formulation of the model or the data. Typically, the problem is caused by missing constraints or the wrong signs on the coefficients.

Note that unboundedness is often diagnosed by presolve. For example:. Here the objective coefficients, constraint coefficients, and right—hand side values range between 0. We say that the model is badly scaled. During the optimization process, the Optimizer must perform many calculations involving subtraction and division of quantities derived from the constraints and the objective function.

When these calculations are carried out with values differing greatly in magnitude, the finite precision of computer arithmetic and the fixed tolerances employed by FICO Xpress result in a build up of rounding errors to a point where the Optimizer can no longer reliably find the optimal solution. To minimize undesirable effects, when formulating your problem try to choose units or equivalently scale your problem so that objective coefficients and matrix elements do not range by more than 10 6 , and the right—hand side and non—infinite bound values do not exceed 10 6.

One common problem is the use of large finite bound values to represent infinite bounds i. College Algebra. Numerical Methods. Statistical Methods. Operation Research. Word Problems. Then, an optimal solution that includes one or more of the slack variables with positive value indicates infeasibility in the corresponding constraint s. Explicitly modeling constraint violations is often recommended in practice since, in reality, many constraints can be violated for a sufficiently high price.

Some LP solvers go beyond just identifying the infeasibility and offer automated approaches for repairing the model. For example:. Dantzig-Wolfe Decomposition , developed by George Dantzig and Philip Wolfe in , is a delayed column generation method for solving large-scale LPs with special structure, specifically a set of coupling constraints and a set of independent submatrices.

To implement a Dantzig-Wolfe Decomposition approach, there are several options:. Performance evaluations of parallel solvers must be interpreted with care. One common measurement is the "speedup" defined as the time for solution using a single processor divided by the time using multiple processors.

A speedup close to the number of processors is ideal in some sense, but it is only a relative measure. The greatest speedups tend to be achieved by the least efficient codes, and especially by those that fail to take advantage of the sparsity predominance of zero coefficients in the constraints.

For problems having thousands of constraints, a sparse single-processor code will tend to be faster than a non-sparse multiprocessor code running on current-day hardware. Most LP textbooks describe how to do sensitivity analysis within the context of the simplex method tableau. The questions of interest are for what range of values of the objective function coefficients or right-hand side values does the solution remain optimal?

Post-optimality analysis covers the cases when feasibility is affected -- by changes in the right-hand side values or by adding a new constraint -- or when optimality is affected -- by changes in the objective function coefficients or by adding a new variable; the choice of algorithm to obtain a new solution efficiently depends on the modifications made. Most LP solvers have features to calculate sensitivity information for the objective function and the constraints, and most LP solvers have built-in procedures for determining the best algorithm primal simplex, dual simplex, generalized simplex for re-solving after modifying the problem.

Note that this theory applies only to linear programming. The development of duality theory and sensitivity analysis for mixed integer programming has not received much attention since the s and the s. For a starting point in learning more about integer programming duality, see the following references:. Interior-point-based LP solvers have their own strategies for selecting a starting point that is within the interior of the feasible region.

Some codes provide an option for loading a user-supplied starting point. Cycling was originally thought to be rare in practice, however, it has been observed particularly in solving highly-structured LPs and in solving LPs that arise as relaxations of integer programming problems. From a theoretical point of view, a number of pivot rules can be implemented that prevent cycling from occurring, for example, the lexicographic pivot rule and Bland's smallest subscript pivot rule.

These rules are not necessarily compatible with practical implementations of the simplex method. Instead, most implementations of the simplex method use a variety of pricing strategies and other features to guard against cycling.

Should you find that your model is affected by cycling, you can try an alternate algorithm as a first step. Connect and share knowledge within a single location that is structured and easy to search. Problem may be infeasible. A non-linear solver will typically give you a local optimum, and the path to get to the solution is a very important part of finding a "better" local optimum. When you tried with another point, you found a feasible solution, and that is the proof that your problem is feasible.

Now, in finding the global optimum instead of a local optimum, this is a little bit harder. One way to find out is to check if your problem is convex. If it is, it means that there will only be one local optimum, and that this local optimum is the global optimum. This can be done mathematically. If you found that your problem is not convex, then you can try to prove that there are few local optimums and that they can be found easily with good starting points.

Finally, if this can't be done, you should consider more advanced techniques, all with their pros and cons. For example, you can try to generate a set of starting solutions to make sure that you cover the whole feasible domain of your problem. Another one would be to use meta-heuristics methods to help you find a better starting solution. Also, I am sure that Ipopt have some tools to help tackling this problem of finding a good starting solution that improves the resulting local optimum.

Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Collectives on Stack Overflow. Learn more. How to know if the optimization problem is infeasible or not? Pyomo Warning: Problem may be infeasible Ask Question.