Estimating NPV for economic projects can result in multiple solutions - see math explanations of this index.

Dear Antonio Bolufé-Röhler,

Here's an article I randomly spotted during a google search

K. Koper and M. Wysession, “Multimodal function optimization with a

niching genetic algorithm: A seis-mological example, Bulletin of the Seismological Society of America vol. 89, pp. 978-988, 1999.

In addition, Swagatam Das and Ponnuthurai Nagaratnam Suganthan have listed a series of real-world multimodal optimization problems for the CEC 2011 Competition on Testing Evolutionary Algorithms on Real World Optimization Problems. The technical report comes with a detailed list of references.

http://sci2s.ugr.es/EAMHCO/srcnew/RealProblemsTech-Rep.pdf

Fitting the largest diamond within a raw stone is a nonlinear optimisation problem with multiple local optima. The problem is straightforward to define and can involve convex shapes but is still difficult to find a global optimum for.

http://www.orsnz.org.nz/conf37/Papers/Viswambharan.pdf

In this paper (https://www.researchgate.net/publication/225158772_Optimization_in_Fractal_and_Fractured_Landscapes_Using_Locust_Swarms?ev=prf_pub), we called the search space for a real-world problem "fractured" and developed tests to show its similarity to a benchmark function with a fractal search space.

To describe in more detail, the angle of incidence for an energy beam is the control  variable.  As this angle changes (in continuous space), it reaches a critical angle at which the wave form goes from reflection to refraction.  Due to this discontinuity in the output variables, the fitness landscape is highly multi-modal.

Conference Paper Optimization in Fractal and Fractured Landscapes Using Locust Swarms

The political districting problem is multimodal (and NP)

Multi-objective problem is actually life problem because any life problem has more than objective .For example , engineering problem , economic problems, and military problem ect....

Multi-objective problem is actually life problem because any life problem has more than objective .For example , engineering problem , economic problems, and military problem ect....

Here is a case where the demand is clearly bi-modal, and where handling the bi-modality is criticial. And this is not caused by multiple objectives, but by fashion.

Hajnalka Vaagen and Stein W. Wallace, Product variety arising from hedging in the fashion supply chains. International Journal on Production Economics 114 (2008) 431--455.

Paper: Optimal placement of piles in real grillages: experimental comparison of optimization algorithms, R.Belevicius, S.Ivanikovas, ... . The link to the programs in Fortran for solution of direct problem is included.

You will be able to compare your algorithm with 5 mentioned in this article.

My example is quite simple and convincing.  Suppose you have samples of a purely sinusoidal signal, irregularly taken.  Trying to guess the original frequency will produce equally good results for some f, 2f, 3f, etc.

Travelling Salesman Problem(TSP) and most of the scheduling problems are multi-modal.

All meta-heuristics (and exact algorithms for global optimization) are "specifically designed" for multi-modal functions... In any case, I guess you are looking for some NLP function coming from a real-world application. Try minimizing (in all three variables) the expected long-run cost of a single-echelon inventory control system operating a periodic review (s,S,T) policy under Poisson demands (the case is very very real, and you can "find" it in almost any retail store that is trying seriously to control its inventories: the assumptions on penalties for backorders is another issue though).

References:

1. A.G. Lagodimos, I.T. Christou, K. Skouri, Computing Globally Optimal (s,S,T) Inventory Policies, OMEGA: The International Journal of Management Science, vol. 40, no. 5, Oct. 2012, pp. 660-671

2. I.T. Christou, Quantitative Methods in Supply Chain Management, Springer, 2012.

3. Y-L Chang, K. Dessai, WinQSB: decision support software for MS/OM. Wiley, 2003.

There are several multi-objective problems in construction management branch. Time , cost, and resource management are always the three goals for researchers in this field. You can find that at scheduling of repetitive projects as example

F.A. Agrama, (2011) “Linear Projects scheduling Using Spreadsheets Features”, Alexandria Engineering Journal, 50(2), pp. 179-185.

F.A. Agrama, (2012) “Multi-objective Genetic Optimization of Linear Construction Projects”, Housing and Building National Research Center Journal, 8, pp. 144-151.

F.A. Agrama, (2014) “Multi-objective Genetic Optimization for Scheduling a Multi-Storey Building”, Automation in Construction Journal, 44, pp. 119-128.

T. Hegazy, and N. Wassef, (2001), “Cost Optimization in Projects with Repetitive Nonserial Activities”, Construction Engineering and Management Journal, ASCE, 127(3), pp. 183-191.

T. Hegazy and E. Kamarah, (2008) "Efficient Repetitive Scheduling for High-Rise Construction”, Journal of Construction Engineering and Management ASCE,134(4), pp. 253–264.

Have you checked this reference: Gould, N.I.M., Orban, D., Toint, Ph.L.: CUTEr (and sifdec), a constrained and unconstrained testing environment, revised. ACM Trans. Math. Softw. 29, 373–394 (2003)?. Maybe there are some problems of your interest there.

Dear Ioannis,

I agree with you that all global optimization methods have implicit mechanisms to escape local optima and thus dealing with multi-modality. However, I also believe that many of these methods were designed to perform well on both, unimodal and multi-modal problems. Furthermore, some heuristics such as Particle Swarm Optimization were initially tested (and designed) using unimodal functions. This could be the reason why initially they used a global topology and later recommended to use a ring (local) topology – “Defining a standard for PSO” paper.

I also believe this is the reason why so many explicit methods for increasing diversification and exploration exist (e.g. niching methods). But in most heuristics these are ad-hoc techniques. Conceived after the design of the (meta)heuristic, or the deterministic algorithm. In our heuristic we took these considerations into account from the beginning. That’s why I stated that it was specifically designed for multi-modal optimization.

Thank you very much for your answer (and all the useful references).

Cheers, Antonio

CEC 2011 problem and matlab code

Using the AHP method for optimal crop modelling.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2633076

Curve and surface fitting problems are, in general, multimodal.

We are working on Business Intelligence solutions developing collaborative platform for self-service analysis, KPI prediction.

In template matching (computer visión problema) is a task of multi-modal nature