This chapter introduces basic models for mixed-integer linear programming. It starts with multiple-choice constraints, and implications that are formulated as inequalities with 0-1 variables. Next, constraints with continuous variables are introduced, such as discontinuous domains and cost functions with fixed charges, which are formulated as linear mixed-integer constraints. Finally, the chapter closes by introducing classic OR problems, the assignment problem, facility location problem, knapsack problem, set covering problem and the traveling salesman problem, all of which are formulated as linear integer and mixed-integer linear programming models.

This chapter addresses the decomposition of MILP optimization problems that involve complicating constraints. It is shown that, by dualizing the complicating constraints, one can derive the Lagrangean relaxation that yields a lower bound to the optimal solution. It is also shown that, by duplicating variables, one can dualize the corresponding equalities yielding the Lagrangean decomposition method that can predict stronger lower bounds than the Lagrangean relaxationn. The steps involved in this decomposition method are described, and can be exended to NLP and MINLP problems.

This chapter first introduces basic concepts in nonlinear optimization, especially feasible regions and convexity conditions. Sufficient conditions are provided for both convex regions and convex functions. Next, optimality conditions are presented for unconstrained optimization problems (stationary conditions), and constrained problems with equality constraints (stationary condition of Lagrange function) and with inequality constraints (Fritz-John Theorem).The chapter concludes with nonlinear optimization with equality and equalityconstraints that lead to the Karush–Kuhn–Tucker conditions. Finally, an active set strategy is introduced for the solution of small nonlinear programming problems, and is illustrated with a small example.

This chapter addresses the global optimization of nonconvex NLP and MINLP optimization problems. The use of convexification transformations is first introduced that allow us to transform a nonconvex NLP into a convex NLP. This is illustrated with geometric programming problems that involve posynomals, and that can be convexified with exponential transformations. We consider next the more general solution approach that relies on the use of convex envelopes that can predict rigorous lower bounds to the global optimum, and which are used in conjunction with a spatial branch and bound method. The case of bilinear NLP problems is addressed as a specific example for which the McCormick convex envelopes are derived. The application of the spatial branch and bound search, coupled with McCormick envelopes, is illustrated with a small example. The software BARON, ANTIGONE, and SCIOP are briefly described.

This chapter addresses the solution of nonlinear programming (NLP)problemsthrough algorithms whose objective is to find a point satisfying the Karush–Kuhn–Tucker conditions through different applications of Newton's method. The algorithms considered include successive-quadratic programming, reduced-gradient method and interior-point method. The basic assumptions behind each method are stated and used to derive the major steps involved in these algoritms. We make brief reference to optimization software including SNOPT, MINOS, CONOPT, IPOPT and KNITRO. Finally, general guidelines are given how to formulate good NLP models.

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.

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.