The representation of complex functions frequently requires the use of infinite series expansions. The best known are Taylor and Laurent series, which represent analytic functions in appropriate domains. Applications often require that we manipulate series by termwise differentiation and integration. These operations may be substantiated by employing the notion of uniform convergence. Series expansions break down at points or curves where the represented function is not analytic. Such locations are termed singular points or singularities of the function. The study of the singularities of analytic functions is vitally important in many applications including contour integration, differential equations in the complex plane, and conformal mappings.

This chapter first describes general approaches for anticipating uncertainty in optimization models. The strategies include optimizing the expected value, minimax stategy, chance-constrained,two-stage and multistage programming, and robust optimization. The chapter focuses on the solution of two-stage stochastic MILP programming problems in which 0-1 variables are present in stage-1 decisions. The discretization of the uncertain parameters is described, which gives rise to scenario trees. We then present the extended MILP formulation that explicitly considers all possible scenarios. Since this problem can become too large, the Benders decomposition method (also known as the L-shaped method )is introduced, in which a master MILP problem is defined through duality in order to predict new integer values for stage-1 decisions, as well as a lower bound. The extension to multistage programming problems is also briefly discussed, as well as a brief reference to robust optmization in which the robust counterpart is derived.

Having introduced mixed-integer linear programming (MILP)models in Chapter 6 using somewhat intuitive arguments, this chapter shows that MILP models can be systematically derived using concepts of propositional logic.The chapter introduces the conjunctive normal form (CNF) as a logic form that can be used as a basis to readily formulate linear constraints with 0-1 variables. Steps are described that are required to transform logic propositions into CNF form. Next the concept of disjunctions is introduced, showing that these can be formulated as MILP constraints either with big-M formulation or with the hull reformulation. It is also shown that the latter leads to strong LP relaxations.

The circumgalactic medium (CGM) is the gas that lies outside the main stellar distribution of a galaxy, but inside its virial radius. The first part of our own galaxy’s CGM to be discovered was a population of high-velocity clouds, discovered through the 21 cm emission of their neutral hydrogen. The high-velocity clouds, however, are embedded within hotter components of the CGM, with temperatures ranging from 104 K to 106 K. These hotter components can be detected through absorption and emission lines of ionized metals such as oxygen. The intracluster medium (ICM) is the gas that lies inside the virial radius of a cluster of galaxies, but which is not associated with any individual galaxy. The ICM can be detected and studied through its free--free emission, which indicates temperatures as high as 108 K.

This chapter addresses the solution of mixed-integer nonlinear programming (MINLP) problems. The following methods for convex MINLP optimization are described: branch and bound, outer-approximation, generalized Benders decomposition. and extended cutting plane. The last three methods rely on decomposing the MINLP problem into a master MILP model thatpredicts lower bounds and new integer values, and an NLP subproblem that is solved for fixed integer variables yielding an upper bound. It is shown that the MILP master problem of generalized Benders decomposition can be derivedfrom a linear combination of the constraints of the master MILP for outer-approximation yielding a weaker lower bound. The extension of these methods for solving nonconvex MINLP problems is discussed, as well as brief reference to software such as SBB, DICOPT, and α-ECP.

This chapter addresses the problem of establishing the feasibilty of a set of constraints given that recourse variables are involved, and that the uncertainty set is specified, typically through lower and upper bounds. This problem, denoted as the feasibility test problem, is shown to correspond to a max-min-max optimization problem. It is shown that, under assumptions of convexity, the problem can be simplified through vertex seaches in the parameter set. It is also shown that the feasibility test problem can be reformulated asa bilevel optimization problem in which the KKT conditions in the inner problem can be reformulated through mixed-integer constraints. It is shown that this MINLP has the capability of predicting nonvertex solutions. The feasibility test is then extended to the feasibility index problem that determines the actual parameter range that is feasible. The concept of one-dimensional convexity is introduced to provide sufficient conditions for the validity of vertex searches. The example of a heat exchanger network is used to illustrate the mathematical formulations.

This chapter first presents basic theoretical concepts of linear programming (LP) problems. These include convexity, solution at extreme points or vertices, and charcterization of these through system of equations expressed in terms of basic and nonbasic variables. The KKT conditions are the applied to identify optimal vertex solutions. These theoretical concepts are then applied to derive the Simplex algorithm, which is introduced as an exchange algorithm between basic and nonbasic variables so as to verify optimality at a given vertex, and ensure feasible steps. A small numerical example is presented to illustrate the steps of the Simplex algorithm. Finally, a brief discussion on software such as CPLEX, GUROBI, and XPRESS is also presented.

A large number of problems arising in fluid mechanics, electrostatics, heat conduction, and many other physical situations, can be mathematically formulated in terms of Laplace’s equation (see also the discussion in Section 2.1).

The hot ionized medium (HIM) represents gas at T ∼ 106 K and n ∼ 0.004 cm−3. It constitutes gas that has been shock-heated by supernova explosions, and which has not yet had time to cool by free--free emission. The properties of a spherically expanding shock front are described by the Sedov–Taylor solution; when radiative losses from the post-shock gas are large, the expanding supernova remnant transitions to the snowplow solution. The hot gas inside a supernova-blown bubble is in collisional ionization equilibrium, which permits a calculation of the ionization state of each element as a function of temperature. Emission lines from ionized iron and absorption lines of ionized oxygen (seen in absorption toward hot white dwarfs) provide information about the density and temperature of the hot gas in the Local Bubble within which the Sun lies.

This chapter provides first an introduction an types of optimization problems that arise in different areas of process systems engineering. It then provides a general classification of optimization problems: linear and mixed-integer linear programming, nonlinear and mixed-integer nonlinear programming, generalized disjunctive programming, decomposition methods, stochastic programming, and flexibility analysis. Finally it reviews the outline of the book through the different chapters.