29 Extended Frontiers in Optimization Techniques Sergiy Butenko and Panos M Pardalos 29.1 Recent Progress in Optimization Techniques Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been devel- oped, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound (Floudas and Pardalos 2002; Pardalos and Resende 2002). At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the inter- disciplinary nature of the field. Optimization today is a basic research tool in all areas of engineering, medicine and the sciences. The decision making tools based on optimization procedures are successfully applied in a wide range of practical problems arising in virtually any sphere of human activities, including biomedi- cine, energy management, aerospace research, telecommunications and finance. In this chapter we will briefly discuss the current developments and emerging chal- lenges in optimization techniques and their applications. The problems of finding the ``best’’ and the ``worst’’ have always been of a great interest. For example, given n sites, what is fastest way to visit all of them consecutively? In manufacturing, how should one cut plates of a material so that the waste is minimized? Some of the first optimization problems were solved in ancient Greece and are regarded among the most significant discoveries of that time. In the first century A.D., the Alexandrian mathematician Heron solved the problem of finding the shortest path between two points by way of the mirror. This result, also known as the Heron’s theorem of the light ray, can be viewed as the origin of the theory of geometrical optics. The problem of finding extreme values gained a special importance in the seventeenth century when it served as one of motivations in the invention of differential calculus. The soon after devel- oped calculus of variations and the theory of stationary values lie in the foundation of the modern mathematical theory of optimization.