# Main idea¶

Goal programming is used to handle an optimization problem with multiple goals of the form

$\begin{split}\text{minimise }& f_i(x), \\ \text{subject to }& g_{ij}(x) \leq 0 \text{ for } j=1,...,m_i,\end{split}$

for $$i=1,...,N$$, with $$N$$ the total number of goals.

Each goal $$i$$ has a weight $$\omega_i$$ and a priority $$p_i$$. Goals with the same priority are converted into one goal. For example, if multiple goals have priority 30, then the resulting minimization problem for priority 30 would look like

$\begin{split}\text{minimise }& \sum_{i:p_i=30} \omega_i f_i(x), \\ \text{subject to }& g_{ij}(x) \leq0 \text{ for }j=1,...,m_i, i: p_i=30.\end{split}$

The reduced set of goals can be rewritten in the form

$\begin{split}\text{minimise }& \sum_{i=1}^{n_p} \omega_{pi} f_{pi}(x), \\ \text{subject to }& g_{pj}(x) \leq 0, \text{ for } j=1,...,m_p,\end{split}$

Problems with different priorities are solved from lowest to highest priority. The results from previous priorities are used as constraints in the goal of the next priority to enforce that the minimal values of the previous priorities remain minimal. For example, suppose we have goals with priorities 10, 20, and 30. We first solve the minimisation problem for priority 10 and find a solution $$x_{10}^*$$. For the next goal, we add the constraints $$f_{10,i}(x) \leq f_{10,i}(x_{10}^*)$$ or, equivalently, $$f_{10,i}(x) - f_{10,i}(x_{10}^*) \leq 0$$, for $$i=1,2,\dots,n_{10}$$. The resulting problem for priority 20 can thus be written as

$\begin{split}\text{minimise }& \sum_{i=1}^{n_{20}} f_{20,i}(x), \\ \text{subject to }& g_{20,j}(x) \leq 0 \text{ for } j=1,...,m_{20}, \\ \text{and }& f_{10,i}(x) \leq f_{10,i}(x_{10}^*) \text{ for } i=1,...,n_{10}.\end{split}$

For priority 30, we then do something similar and add the constraints $$f_{10}(x) \leq f_{10,i}(x_{10}^*)$$ and $$f_{20}(x) \leq f_{20,i}(x_{20}^*)$$. In general, the extended problem for priority $$p$$ can be written as

$\begin{split}\text{minimise }& \sum_{i=1}^{n_p} f_{pi}(x), \\ \text{subject to }& g_{qj}(x) \leq 0, \text{ for } j=1,...,m_{p}, q \leq p, \\ \text{and }& f_{qi}(x) \leq f_{qi}(x_{q}^*), \text{ for } i=1,...,n_q, q < p.\end{split}$

This is the default way to convert results of a previous optimisation problem to a constraint for the next optimisation problem. Details on converting goals to constraints can be found in Converting goals to constraints.

There are several options for implementing goal programming. These options are set with rtctools.optimization.goal_programming_mixin.GoalProgrammingMixin.goal_programming_options().

class rtctools.optimization.goal_programming_mixin.GoalProgrammingMixin(**kwargs)[source]

Bases: rtctools.optimization.goal_programming_mixin_base._GoalProgrammingMixinBase

goal_programming_options() → Dict[str, Union[float, bool]][source]

Returns a dictionary of options controlling the goal programming process.

Option Type Default value
violation_relaxation float 0.0
constraint_relaxation float 0.0
mu_reinit bool True
fix_minimized_values bool True/False
check_monotonicity bool True
equality_threshold float 1e-8
interior_distance float 1e-6
scale_by_problem_size bool False
keep_soft_constraints bool False

Before turning a soft constraint of the goal programming algorithm into a hard constraint, the violation variable (also known as epsilon) of each goal is relaxed with the violation_relaxation. Use of this option is normally not required.

When turning a soft constraint of the goal programming algorithm into a hard constraint, the constraint is relaxed with constraint_relaxation. Use of this option is normally not required. Note that:

1. Minimization goals do not get constraint_relaxation applied when fix_minimized_values is True.
2. Because of the constraints it generates, when keep_soft_constraints is True, the option fix_minimized_values needs to be set to False for the constraint_relaxation to be applied at all.

A goal is considered to be violated if the violation, scaled between 0 and 1, is greater than the specified tolerance. Violated goals are fixed. Use of this option is normally not required.

When using the default solver (IPOPT), its barrier parameter mu is normally re-initialized at every iteration of the goal programming algorithm, unless mu_reinit is set to False. Use of this option is normally not required.

If fix_minimized_values is set to True, goal functions will be set to equal their optimized values in optimization problems generated during subsequent priorities. Otherwise, only an upper bound will be set. Use of this option is normally not required. Note that a non-zero goal relaxation overrules this option; a non-zero relaxation will always result in only an upper bound being set. Also note that the use of this option may add non-convex constraints to the optimization problem. The default value for this parameter is True for the default solvers IPOPT/BONMIN. If any other solver is used, the default value is False.

If check_monotonicity is set to True, then it will be checked whether goals with the same function key form a monotonically decreasing sequence with regards to the target interval.

The option equality_threshold controls when a two-sided inequality constraint is folded into an equality constraint.

The option interior_distance controls the distance from the scaled target bounds, starting from which the function value is considered to lie in the interior of the target space.

If scale_by_problem_size is set to True, the objective (i.e. the sum of the violation variables) will be divided by the number of goals, and the path objective will be divided by the number of path goals and the number of active time steps (per goal). This will make sure the objectives are always in the range [0, 1], at the cost of solving each goal/time step less accurately.

The option keep_soft_constraints controls how the epsilon variables introduced in the target goals are dealt with in subsequent priorities. If keep_soft_constraints is set to False, each epsilon is replaced by its computed value and those are used to derive a new set of constraints. If keep_soft_constraints is set to True, the epsilons are kept as variables and the constraints are not modified. To ensure the goal programming philosophy, i.e., Pareto optimality, a single constraint is added to enforce that the objective function must always be at most the objective value. This method allows for a larger solution space, at the cost of having a (possibly) more complex optimization problem. Indeed, more variables are kept around throughout the optimization and any objective function is turned into a constraint for the subsequent priorities (while in the False option this was the case only for the function of minimization goals).

Returns: A dictionary of goal programming options.
goals() → List[rtctools.optimization.goal_programming_mixin_base.Goal]

User problem returns list of Goal objects.

Returns: A list of goals.
path_goals() → List[rtctools.optimization.goal_programming_mixin_base.Goal]

User problem returns list of path Goal objects.

Returns: A list of path goals.
priority_completed(priority: int) → None

Called after optimization for goals of certain priority is completed.

Parameters: priority – The priority level that was completed.
priority_started(priority: int) → None

Called when optimization for goals of certain priority is started.

Parameters: priority – The priority level that was started.
class rtctools.optimization.single_pass_goal_programming_mixin.SinglePassGoalProgrammingMixin(**kwargs)[source]

Bases: rtctools.optimization.goal_programming_mixin_base._GoalProgrammingMixinBase

Unlike GoalProgrammingMixin, this mixin will call transcribe() only once per call to optimize(), and not $$N$$ times for $$N$$ priorities. It works similar to how keep_soft_constraints = True works for GoalProgrammingMixin, while avoiding the repeated calls to transcribe the problem.

This mixin can work in one of two ways. What is shared between them is that all violation variables of all goals are generated once at the beginning, such that the state vector is exactly the same for all priorities. They also share that all goal constraints are added from the start. How they differ is in how they handle/append the constraints on the objective of previous priorities:

1. At priority $$i$$ the constraints are the same as the ones at priority $$i - 1$$ with the addition of the objective constraint related to priority $$i - 1$$. This is the default method.
2. All objective constraints are added at the start. The objective constraints will have bound of $$[-\inf, \inf]$$ at the start, to be updated after each priority finishes.

There is a special qpsol alternative available CachingQPSol, that will avoid recalculations on constraints that were already there in previous priorities. This works for both options outlined above, because the assumptions of CachingQPSol are that:

1. The state vector does not change
2. Any new constraints are appended at the end

Note

Just like GoalProgrammingMixin, objective constraints are only added on the goal objectives, not on any custom user objective.

goal_programming_options() → Dict[str, Union[float, bool]][source]

Returns a dictionary of options controlling the goal programming process.

Option Type Default value
constraint_relaxation float 0.0
mu_reinit bool True
fix_minimized_values bool True/False
check_monotonicity bool True
equality_threshold float 1e-8
scale_by_problem_size bool False

When a priority’s objective is turned into a hard constraint, the constraint is relaxed with constraint_relaxation. Use of this option is normally not required. Note that:

When using the default solver (IPOPT), its barrier parameter mu is normally re-initialized at every iteration of the goal programming algorithm, unless mu_reinit is set to False. Use of this option is normally not required.

If fix_minimized_values is set to True, goal functions will be set to equal their optimized values in optimization problems generated during subsequent priorities. Otherwise, only an upper bound will be set. Use of this option is normally not required. Note that the use of this option may add non-convex constraints to the optimization problem. The default value for this parameter is True for the default solvers IPOPT/BONMIN. If any other solver is used, the default value is False.

If check_monotonicity is set to True, then it will be checked whether goals with the same function key form a monotonically decreasing sequence with regards to the target interval.

The option equality_threshold controls when a two-sided inequality constraint is folded into an equality constraint.

If scale_by_problem_size is set to True, the objective (i.e. the sum of the violation variables) will be divided by the number of goals, and the path objective will be divided by the number of path goals and the number of active time steps (per goal). This will make sure the objectives are always in the range [0, 1], at the cost of solving each goal/time step less accurately.

Returns: A dictionary of goal programming options.
goals() → List[rtctools.optimization.goal_programming_mixin_base.Goal]

User problem returns list of Goal objects.

Returns: A list of goals.
path_goals() → List[rtctools.optimization.goal_programming_mixin_base.Goal]

User problem returns list of path Goal objects.

Returns: A list of path goals.
priority_completed(priority: int) → None

Called after optimization for goals of certain priority is completed.

Parameters: priority – The priority level that was completed.
priority_started(priority: int) → None

Called when optimization for goals of certain priority is started.

Parameters: priority – The priority level that was started.
class rtctools.optimization.single_pass_goal_programming_mixin.CachingQPSol[source]

Bases: object

Alternative to ca.qpsol() that caches the Jacobian between calls.

Typical usage would be something like: