178 Beam-based Correction and Optimization for Accelerators
7.2 REVIEW OF OPTIMIZATION ALGORITHMS
Function optimization is an extensively researched area. There are numerous
optimization algorithms, which cannot be thoroughly reviewed in the scope
of this book. Here we only intend to discuss some well known algorithms that
are potentially applicable to online optimization.
The optimization problem defined in Eq. (7.2) has constraints on the input
variables, while many of the classical optimization algorithms are for unconstrained cases. However, the convergence path from the initial solution to the
optimum is often not affected by the parameter boundary. The parameter
ranges may serve as a sanity check, as well as a safety insurance. The simple
constraints on the parameter ranges in Eq. (7.2) are easy to enforce. For example, when the trial solution is outside the boundary, the objective function
is not evaluated on the machine; instead, a large function value is assigned according to its distance from the boundary. Since most optimization algorithms
will steer away from areas with large function values, the trial solution will
likely move back into the valid parameter space. A not-a-number (NaN) value
may also be assigned when the trial solution is outside of the valid parameter
range, although in this case the algorithms have to be implemented to handle
the NaN value properly. A simpler approach would be for the algorithm to
stop when the boundary is reached.
Here we will consider general unconstrained optimization algorithms for
multi-variable, nonlinear functions. The traditional optimization algorithms
in this area can be classified into two groups, deterministic and stochastic
algorithms. The development of machine learning has introduced new techniques to the optimization field, which may be characterized as model-based
algorithms. For noise free functions, the convergence path from any initial
point is fixed for the deterministic algorithms. On the contrary, the stochastic
algorithms have different paths every time as they employ some randomness in
the choice of the trial solutions. Model-based algorithms build models with the
measurement data and use the models to guide the search for the optimum.
7.2.1 Deterministic optimization algorithms
The deterministic algorithms may be divided into two camps,
• gradient-based methods, and
• gradient-free methods.
7.2.1.1 Gradient-based methods
The gradient-based methods include those that require the calculation of the
derivatives of the objective function. The objective function in the vicinity of
7.2 REVIEW OF OPTIMIZATION ALGORITHMS
Function optimization is an extensively researched area. There are numerous
optimization algorithms, which cannot be thoroughly reviewed in the scope
of this book. Here we only intend to discuss some well known algorithms that
are potentially applicable to online optimization.
The optimization problem defined in Eq. (7.2) has constraints on the input
variables, while many of the classical optimization algorithms are for unconstrained cases. However, the convergence path from the initial solution to the
optimum is often not affected by the parameter boundary. The parameter
ranges may serve as a sanity check, as well as a safety insurance. The simple
constraints on the parameter ranges in Eq. (7.2) are easy to enforce. For example, when the trial solution is outside the boundary, the objective function
is not evaluated on the machine; instead, a large function value is assigned according to its distance from the boundary. Since most optimization algorithms
will steer away from areas with large function values, the trial solution will
likely move back into the valid parameter space. A not-a-number (NaN) value
may also be assigned when the trial solution is outside of the valid parameter
range, although in this case the algorithms have to be implemented to handle
the NaN value properly. A simpler approach would be for the algorithm to
stop when the boundary is reached.
Here we will consider general unconstrained optimization algorithms for
multi-variable, nonlinear functions. The traditional optimization algorithms
in this area can be classified into two groups, deterministic and stochastic
algorithms. The development of machine learning has introduced new techniques to the optimization field, which may be characterized as model-based
algorithms. For noise free functions, the convergence path from any initial
point is fixed for the deterministic algorithms. On the contrary, the stochastic
algorithms have different paths every time as they employ some randomness in
the choice of the trial solutions. Model-based algorithms build models with the
measurement data and use the models to guide the search for the optimum.
7.2.1 Deterministic optimization algorithms
The deterministic algorithms may be divided into two camps,
• gradient-based methods, and
• gradient-free methods.
7.2.1.1 Gradient-based methods
The gradient-based methods include those that require the calculation of the
derivatives of the objective function. The objective function in the vicinity of