Table 2.2: Instances for which the feasible solution found is also the best-know solution ex1222 nuclear24b nuclearvd st e27
ex1266a nuclear24 nuclearve st e32 feedtray2 nuclear25a nuclearvf st miqp1 nuclear14a nuclear25 nvs03 st test1 nuclear14b nuclearva nvs15 st test5 nuclear14 nuclearvb prob02 tln2 nuclear24a nuclearvc prob03 tltr
Table 2.3: Instances for which no feasible solution was found within the time limit deb10 fo9 ar25 1 nuclear49a tln12
ex1252 gasnet nuclear49b tls12
fo8 ar25 1 lop97ic nuclear49 tls6 fo8 ar3 1 lop97icx product2 tls7 fo9 ar2 1 nuclear10a space960
any MINLP feasible solution are 19, see Table 2.3. The remaining 16 instances encounter some problems during the execution (see Table 2.4).
Table 2.4: Instances with problems during the execution 4stufen ex1252a risk2b super3
beuster nuclear104 st e40 super3t cecil 13 nuclear10b super1 waste
eg int s pump super2 windfac
2.5
Conclusions
In this chapter we presented a Feasibility Pump (FP) algorithm aimed at solving non-convex Mixed Integer Non-Linear Programming problems. The proposed algorithm is tailored to limit the impact of the non-convexities in the MINLPs. These difficulties aare extensively discussed. In preliminary results we show the algorithm behaves well with general problems presenting computational results on instances taken from MINLPLib.
Chapter 3
A Global Optimization Method for
a Class of Non-Convex MINLP
Problems
13.1
Introduction
The global solution of practical instances of Mixed Integer Non-Linear Programming (MINLP) problems has been considered for some decades. Over a considerable period of time, tech- nology for the global optimization of convex MINLP (i.e. the continuous relaxation of the problem is a convex program) had matured (see, for example, [45, 108, 20]), and rather re- cently there has been considerable success in the realm of global optimization of non-convex MINLP (see, for example, [111, 99, 84, 14]).
Global optimization algorithms, e.g., spatial Branch-and-Bound approaches like those implemented in codes like BARON [111] and Couenne [14], have had substantial success in tackling complicated, but generally small scale, non-convex MINLPs (i.e., mixed-integer non- linear programs having non-convex continuous relaxations). Because they are aimed at a rather general class of problems, the possibility remains that larger instances from a simpler class may be amenable to a simpler approach.
We focus on separable MINLPs, that is where the objective and constraint functions are sums of univariate functions. There are many problems that are already in such a form, or can be brought into such a form via some simple substitutions. In fact, the first step in spatial Branch-and-Bound is to bring problems into nearly such a form. For our purposes, we shift that burden back to the modeler. We have developed a simple algorithm, implemented at the level of a modeling language (in our case AMPL, see [55]), to attack such separable prob- lems. First, we identify subintervals of convexity and concavity for the univariate functions using external calls to MATLAB [91]. With such an identification at hand, we develop a convex MINLP relaxation of the problem (i.e., as a mixed-integer non-linear programs having a convex continuous relaxations). Our convex MINLP relaxation differs from those typically
1This is a working paper with Jon Lee and Andreas W¨achter (Department of Mathematical Sciences, IBM
T.J. Watson Research Center, Yorktown Heights, NY). This work was partially developed when the author of the thesis was visiting the IBM T.J. Watson Research Center and their support is gratefully acknowledged.
employed in spatial Branch-and-Bound; rather than relaxing the graph of a univariate func- tion on an interval to an enclosing polygon, we work on each subinterval of convexity and concavity separately, using linear relaxation on only the “concave side” of each function on the subintervals. The subintervals are glued together using binary variables. Next, we employ ideas of spatial Branch-and-Bound, but rather than branching, we repeatedly refine our con- vex MINLP relaxation by modifying it at the modeling level. We attack our convex MINLP relaxation, to get lower bounds on the global minimum, using the code Bonmin [20, 26] as a black-box convex MINLP solver. Next, by fixing the integer variables in the original non- convex MINLP, and then locally solving the associated non-convex NLP relaxation, we get an upper bound on the global minimum, using the code Ipopt [123]. We use the solutions found by Bonmin and Ipopt to guide our choice of further refinements.
We implemented our framework using the modeling language AMPL. In order to obtain all of the information necessary for the execution of the algorithm, external software, specifically the tool for high-level computational analysis MATLAB, the convex MINLP solver Bonmin and the NLP solver Ipopt, are called directly from the AMPL environment. A detailed description of each part and of the entire algorithmic framework is provided in S3.2.
We present computational results in S3.3. Some of the instances used arise from specific applications; in particular, Uncapacitated Facility Location and also Hydro Unit Commitment and Scheduling. We also present computational results on selected instances of GLOBALLib and MINLPLib. We have had modest success in our preliminary computational experiments. In particular, we see very few major iterations occurring, with most of the time is spent in the solution of a small number of convex MINLPs. An advantage of our approach is that further advances in technology for convex MINLP will immediately give us a proportional benefit.