Optimization is everywhere. It is human nature to seek the best option among all that are available. Nature, too, seems to be guided by optimization|many laws of nature have a variational character. Among geometric figures in the plane with a fixed perimeter, the circle has the greatest area. Such isoperimetric problems involving geometric figures date back to ancient Greece. Fermat's principle, discovered in 1629, stating that the tangent line is horizontal at a minimum point, seems to have influenced the development of calculus. The proofs of Rolle's theorem and the mean value theorem in calculus use the Weierstrass theorem on the existence of maximizers and minimizers. The introduction
of the brachistochrone problem in 1696 by Johann Bernoulli had a tremendous impact on the development of the calculus of variations and influenced the development of functional analysis. The variational character of laws of mechanics and optics were discovered in the seventeenth and eighteenth centuries. Euler and Lagrange forged the foundations of the calculus of variations in the eighteenth century. In the nineteenth century, Riemann used Dirichlet's principle, which has a variational character, in his investigations in complex analysis. The simplex method for linear programming was discovered shortly after the advent of computers in the 1940s, and influenced the subsequent development of mathematical programming. The emergence of the theory of optimal control in the 1950s was in response to the need for controlling space vehicles and various industrial processes. Today, optimization is a vast subject with many subelds, and it is growing at a rapid pace. Research is proceeding in various directions|advancement of theory, development of new applications and computer codes, and establishment or renewal of ties with many fields in science and industry.
The main focus of this book is optimization in finite-dimensional spaces. The book is suitable as a textbook for a first course in the theory of optimization in finite-dimensional spaces at the graduate level. The book is also suitable for self-study or as a reference book for more advanced readers. An important feature of the book is the inclusion of over two hundred carefully selected exercises as well as a fair number of completely solved examples within the text.