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This paper introduces constructing convex-relaxed programs for nonconvex optimization problems. Branch-and-bound algorithms are convex-relaxation-based techniques. The convex envelopes are important, as they represent the uniformly best convex underestimators for nonconvex polynomials over some region. The reformulation-linearization technique (RLT) generates linear programming (LP) relaxations of a quadratic problem. RLT operates in two steps: a reformulation step and a linearization (or convexification) step. In the reformulation phase, the constraint and bound inequalities are replaced by new numerous pairwise products of the constraints. In the linearization phase, each distinct quadratic term is replaced by a single new RLT variable. This RLT process produces an LP relaxation. The LP-RLT yieds a lower bound on the global minimum. LMI formulations (linear matrix inequalities) have been proposed to treat efficiently with nonconvex sets. An LMI is equivalent to a system of polynomial inequalities. A semialgebraic convex set describes the system. The feasible sets are spectrahedra with curved faces, contrary to the LP case with polyhedra. Successive LMI relaxations of increasing size yield the global optimum. Nonlinear inequalities are converted to an LMI form using Schur complements. Optimizing a nonconvex polynomial is equivalent to the LP over a convex set. Engineering application interests include system analysis, control theory, combinatorial optimization, statistics, and structural design optimization.

Technical terms related to mathematical programming complement and clarify the proposed definitions. Other technical terms chosen relate to the background in which mathematical optimization can take place in mathematics, operations research, statistics, and probabilities. The additional terms used are defined more succinctly. These terms, whose presentation is close to that of the essential terms of mathematical optimization, include a block of information, examples, and graphic illustrations.

The academic literature includes many useful surveys to assess the origin and the circumstances of discovery and to put into perspective the evolution of a particular method. The user may also feel the need to refer to more focused, precise, and illustrated information that shows the value of an approach. The design of this presentation of terms is based on a chosen (nonexhaustive) set of items is user-oriented in view of its own applications. This central part of the book deals with the definition and presentation of entries on the mathematical optimization terminology in alphabetical order. The list aims to be complete in the sense of the contemporary dimensions of optimization. Each simple terminology is the subject of a comparable questioning to know the origin, the definition, mathematical formulation, and illustrations. An information block accompanying each entry includes useful data to facilitate a further study.

Optimization methods and applications represent a vast literature. The intensity and variety of research papers have increased in recent years. This introduction gives a precise historical overview of these developments. The use of evolutionary algorithms has emerged in combinatorial optimization and optimization in continuous or mixed design variables. The reasons are the conventional methods failed before the formal complexity of full-scale applications in real life. Two major types of problems stand out, that is, the decision problems involving a single objective and challenges requiring use of multiple objectives jointly. Classification of optimization problems can be based on the technical characteristics of these problems. However, the classification of methods differs for single and multiobjective optimization. Regarding single-objective problems, a distinction is introduced between direct research methods and methods of descent with derivatives. The classification methods of multiobjective optimization are based on the number of Pareto solutions, as well as the preferences of decision maker.

Diffusion mechanisms in natural sciences and innovation management involve partial differential equations (PDEs). This is due to their spatio-temporal dimensions. Functional semi-discretized PDEs (with lattice spatial structures or time delays) may be even more adapted to real world problems. In the modeling process, PDEs can also formalize behaviors, such as the logistic growth of populations with migration, and the adopters’ dynamics of new products in innovation models. In biology, these events are related to variations in the environment, population densities and overcrowding, migration and spreading of humans, animals, plants and other cells and organisms. In chemical reactions, molecules of different species interact locally and diffuse. In the management of new technologies, the diffusion processes of innovations in the marketplace (e.g., the mobile phone) are a major subject. These innovation diffusion models refer mainly to epidemic models. This contribution introduces that modeling process by using PDEs and reviews the essential features of the dynamics and control in biological, chemical and new technology transfer. This paper is essentially user-oriented with basic nonlinear evolution equations, delay PDEs, several analytical and numerical methods for solving, different solutions, and with the use of mathematical packages, notebooks and codes. The computations are carried out by using the software Wolfram Mathematica®7, and C++ codes.