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.

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.

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.

This chapter addressed the solution of mixed-integer linear (MILP) problems for which first simple methods are introduced, namely exhaustive enumeration of all 0-1 combinations, and solution through relaxation and rounding. The branch and bound method is then formally introduced, identifying major properties such as lower bounding in order to use these tofathom nodes, or to obtain feasibleinteger solutions which yield upper bounds. The concept of cutting planes is also introduced with Gomory's cutting plane. Finally, the combination of branch and bound and cutting planes is discussed. Finally, a brief discussion on software such as CPLEX, GUROBI, and XPRESS is presented.

This chapter first intruduces the idea of process modeling, both as equation oriented and as sequential modular calculations. It then poses the analysis of process flowsheets as a system of nonlinear equations. Newton's method is first introduced for solving the systems of nonlinear equations, highlighting some of its major theoretical properties. Next, the concept of quasi-Newton methods is introduced to approximate the Jacobian matrix in Newton's method. The specific quasi-Newton method presented is Broyden's method for which its derivation is presented.

This chapter introduces constraint programming, which is a modeling framework that can accommodate discrete, Boolean, and continuous variables, and where constraints can be algebraic, or in the form of disjunctions, logic constraints, or global constraints that represent procedures. The main goal in constraint programming is to find feasible solutions to the specified model. The main solution method relies on a tree search that relies on domain reduction and constraint propagation techniques.As an example of these constraints, "edge-finding constraints" for the area of scheduling are presented to illustrate the procedural aspect of the search. A simple example problem is presented to illustrate the tree search used in conjunction with domain reduction and constraint propagation. The software OPL is briefly described.

This chapter extends the generalized disjunctive programming (GDP) model introduced in Chapter 7 to problems involved nonlinear objectives and nonlinear constraints. Relaxations of the GDP problem defined by the big-Mand hull reformulations are introduced, which can be used as a basis to reformulate the GDP problems as an MINLP. For the case of the hull reformulation, an approximation of the perspective function is presented that is exact for the 0-1 integer values. A disjunctive branch and bound method is described that can use as the relaxation the one of the big-M to the hull reformulation. Also, this chapter presents the logic-based outer-approximation method that can be used to solve GDP models for superstructure optimization of process flowsheets.

The chapter focuses on network flow problems, which form a very important part of practical applications.Routing, distribution, and scheduling problems often belong to this category of formulations, while a large number of other optimization problems encountered in diverse areas of applications may contain elements of network flow problems.

Quadratic multidimensional functions play a very important role in the understanding of general nonlinear functions. Convexity of quadratic functions is linked in a natural way from its geometrical definition all the way to the properties of its matrix eigenspectrum.Indeed, to second order expansion, and close to the expansion point, any nonlinear function can be approximated by a quadratic – thus providing a crucial link and understanding of the local behaviour and convexity properties of general functions.

The chapter introduces basic optimization concepts, and motivates the use of optimization models and methods to engineering and scientific practice applications.It establishes key concepts, such as the types of variables, arguments to an optimization problem as continuous, integer and control functions (for optimal control problems).Further, it introduces types of optimization problems according to their formulation (such as multiobjective, bilevel, stochastic optimization problems)

This chapter introduces concepts of norm-1 and infinity norm fitting, both in terms of their own merit as useful fitting techniques, apart from least squares, but also importantly to teach how optimization problems that seem hard to solve (such as by being non-differentiable) can be reformulated effectively into easier ones that can be handled by standard solution methods – in this case by LP solvers.