NIC- M1-CH1-OPTIMIZATION

Overview of Optimization

Summary

• This chapter defines optimization and its basic concepts.

• Provides examples of engineering optimization problem

1.1 Optimization

• Engineers design and operate systems to meet or surpass set goals within constraints.

• Optimization: Organized search for designs and operating modes, determining optimal actions or elements to achieve optimized systems.

• Generally seeks maximum or minimum values of an objective function while adhering to constraints.

• A single-objective optimization model consists of:

  • Objective function: f(X)

  • Decision variables: X = (x1, x2, ..., xN)

  • Constraints: g;(X) < b;

  • Bounds: x(L) ≤ x ≤ x(U)

• Notation:

  • f(x) = the objective function

  • X = a set of decision variables

  • g;(X) = jth constraint

  • b; = constant of the jth constraint

  • m = total number of constraints

  • x(L) = lower bound

  • x(U) = upper bound

1.1.1 Objective Function

• The objective function represents the optimization goal, which can be maximized or minimized by selecting appropriate decision variables.

• Decision variables satisfy constraints that influence the value of the objective function.

• Maximization or minimization problems can be converted by changing the sign of the objective function.

1.1.2 Decision Variables

• Decision Variables: Determine the value of the objective function.

• Can be continuous (e.g., proportions of substances) or discrete (specific values like integers).

• Continuous decision variables form a space in which any value in a range (e.g., [0,1]) can be chosen.

  • Example: Making a mixture with proportions.

• Discrete decision variables take specific integer values.

  • Example: Number of groundwater wells.

• Mixed-type optimization: A mix of binary (0 or 1) and continuous variables; e.g., siting and capacity decisions.

1.1.3 Solutions of an Optimization Problem

• Solutions are expressed in terms of decision variables.

• One decision variable = one-dimensional; multiple variables = N-dimensional.

1.1.4 Decision Space

• The feasible decision space is the set of decision variables that meet constraints.

• Each solution corresponds to an N-vector variable.

• Optimization algorithms search for vectors in the decision space that optimize the objective function.

1.1.5 Constraints or Restrictions

• Two constraint types:

  • Directly restrict decision variables (e.g., x>0).

  • Formulas restrict relationships between variables (e.g., x1 + x2 ≤ b).

• The goal is to find an optimal feasible solution that meets all constraints.

1.1.6 State Variables

• State Variables: Dependent variables whose values change with decision variables.

• Important in models to describe systems being optimized (e.g., hydropower generation).

1.1.7 Local and Global Optima

• A well-defined optimization problem has a defined decision space, where each point corresponds to the objective function value.

• Local Optimum: Best in its neighborhood but may not be globally optimal.

• Global Optimum: Best in the entire decision space.

• There can be multiple global optima in situations like linear programming.

• Decision spaces can be:

  • Single-modal: one local and global optimum.

  • Multimodal: multiple local and global optima.

1.1.8 Near-Optimal Solutions

• A near optimum approximates the global optimum closely but may not be the best due to complexity or computation limits.

• Useful for practical problems where absolute optimality is difficult.

1.1.9 Simulation

• Simulation evaluates state variables to estimate the objective function and constraints with decision variables.

• Involves computational modeling of real-world systems.

1.2 Examples of the Formulation of Various Engineering Optimization Problems

1.2.1 Mechanical Design

• Example: Designing compound gear trains to achieve a specified gear ratio.

• Optimization problem seeks to minimize the error between required and obtained gear ratios.

• Decision variables: Number of teeth on each gear, integer values.

• Minimize objective function: f(x) = (-a)², with constraints on teeth numbers.

1.2.2 Structural Design

• Optimize structural configurations to meet specs while minimizing weight or deflection of structures.

• Example: Two-bar truss design.

• Stresses computed as a function of forces and dimensions.

• Decision variables involved in optimizing the structure's cross-sectional areas and dimensions.

1.2.3 Electrical Engineering Optimization

• Directional Overcurrent Relays (DOCRs) protect transmission systems.

• Ensure reliable relay coordination to isolate faults efficiently.

• Decision variables include time dial settings and plug settings.

• Nonlinear objective function with constraints relating to operating times.

1.2.4 Water Resources Optimization

• Optimizing hydropower generation through reservoir management.

• Storage equations govern water flow and power generation in the dam.

• Decision variables include water release volumes in operational time periods to maximize power output.

1.2.5 Calibration of Hydrologic Models

• Calibration involves minimizing errors between observed and predicted hydrologic values (e.g., flood routing).

• Decision variables relate to parameters of the hydrologic model.

• Objective function focuses on minimizing squared differences over time steps.