Inside Process Optimization: When Reality Becomes Nonlinear and Decisions Become Discrete
In the previous article, we moved from problem formulation to linear programming, the first systematic way to solve structured optimization problems. But real chemical processes rarely behave in straight lines.
Reaction kinetics are nonlinear. Thermodynamics introduces curvature. And many engineering decisions are not continuous at all; they are yes-or-no. This is where optimization becomes both more realistic and more challenging.
Chapter 5 introduces nonlinear programming (NLP), the framework needed when process relationships are inherently nonlinear. This is the natural domain of chemical engineering: reaction rates depend exponentially on temperature, thermodynamic properties follow complex relationships, and process performance emerges from interacting nonlinear effects.
In this setting, optimization is no longer about straight lines and corner points. Instead, we deal with curved feasible regions, gradients, and optimality conditions. Concepts such as Lagrange multipliers provide insight into how constraints influence the solution, while numerical methods are required to actually compute it.
A key realization here is that solutions are not always unique or global — they can be local, depending on where the algorithm starts. This makes both modeling and initialization essential parts of the optimization task.
Expert spotlight: Prof. Johan Grievink . In his contribution, “Balancing Between Context and Contents in (Industrial) Optimization Applications,” Johan Grievink highlights that successful optimization is not just about mathematics. It is about maintaining a balance between the real-world system (context) and the mathematical model and solver (contents). A model that is mathematically elegant but disconnected from reality has little value; equally, a detailed process description without a solvable formulation is ineffective. Optimization works best when these two sides reinforce each other.
This perspective is especially important in nonlinear problems, where model fidelity and numerical behavior are closely intertwined. While nonlinear programming captures realistic process behavior, it still assumes that decisions can vary continuously. In practice, many decisions are discrete. You either install a unit or you don’t. You assign a batch to a reactor or you don’t. You select a process route from a finite set of alternatives.
Chapter 6 introduces integer and mixed-integer programming (IP, MILP, MINLP), which allow optimization models to include these structural and logical decisions. This opens the door to modeling process design, superstructures, scheduling, and supply chain optimization in a realistic way.
However, this added realism comes at a cost. Discrete decisions turn optimization into a combinatorial problem, where the number of possibilities can grow exponentially. Techniques such as branch-and-bound and decomposition methods are therefore essential to systematically explore the solution space.
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Expert perspective: Prof. Ignacio Grossmann . In his contribution, “Evolution of Optimization in Process Systems Engineering (PSE),” Ignacio Grossmann reflects on how optimization methods have developed within chemical engineering. From early linear models in refinery planning to advanced nonlinear and mixed-integer formulations, optimization has become a central tool for designing and operating complex process systems.
His perspective shows that many of the algorithms and modeling approaches used today, particularly for MINLP problems, have been shaped by the PSE community itself. What may appear as abstract mathematical tools are in fact deeply rooted in real industrial challenges.
Chapters 5 and 6 bring optimization into the space where most real engineering problems exist:
Together, they transform optimization from a clean mathematical exercise into a powerful framework for real-world decision-making. This is where the true nature of process optimization becomes visible: not just solving equations, but combining physical insight, mathematical structure, and computational tools.
Next Month
In the next article, we move from single-objective optimization to multi-objective decision-making, where engineers must balance competing goals such as cost, sustainability, safety, and performance.
Because in real engineering, there is rarely a single definition of “optimal.”
Edwin Zondervan
For steady state optimization the most difficult thing is to find all real world boundary conditions and should the objective function represent the company 's overall target or the plant managers goals? Not necessarily the same. For dynamic optimization the most difficult thing is discrete decisions like switching on or off machines. Some hysteresis, a perfect forecast and a penalty function are key.