Alpine._add_multilinear_linking_constraintsMethod
_add_multilinear_linking_constraints(m::Optimizer, λ::Dict)

This internal function adds linking constraints between λ multipliers corresponding to multilinear terms that share more than two variables and are partitioned. For example, suppose we have λ[i], λ[j], and λ[k] where i=(1,2,3), j=(1,2,4), and k=(1,2,5). λ[i] contains all multipliers for the extreme points in the space of (x1,x2,x3). λ[j] contains all multipliers for the extreme points in the space of (x1,x2,x4). λ[k] contains all multipliers for the extreme points in the space of (x1,x2,x5).

Using λ[i], λ[j], or λ[k], we can express multilinear function x1x2. We define a linking variable μ(1,2) that represents the value of x1x2. Linking constraints are μ(1,2) == convex combination expr for x1x2 using λ[i], μ(1,2) == convex combination expr for x1x2 using λ[j], and μ(1,2) == convex combination expr for x1*x2 using λ[k].

Thus, these constraints link between λ[i], λ[j], and λ[k] variables.

Reference: J. Kim, J.P. Richard, M. Tawarmalani, Piecewise Polyhedral Relaxations of Multilinear Optimization, http://www.optimization-online.org/DB_HTML/2022/07/8974.html

Alpine._get_discretization_dictMethod
_get_discretization_dict(m::Optimizer, lbs::Vector{Float64}, ubs::Vector{Float64})

Utility functions to convert bounds vectors to Dictionary based structures that are more suitable for partition operations.

Alpine._get_shared_multilinear_terms_infoFunction
_get_shared_multilinear_terms_info(λ, linking_constraints_degree_limit)

This function checks to see if linking constraints are necessary for a given vector of each multilinear terms and returns the approapriate linking constraints information.

Alpine.add_adaptive_partitionMethod
add_adaptive_partition(m::Optimizer; use_disc::Dict, use_solution::Vector)

A built-in method used to add a new partition on feasible domains of variables chosen for partitioning.

This can be illustrated by the following example. Let the previous iteration's partition vector on variable "x" be given by [0, 3, 7, 9]. And say, the lower bounding solution has a value of 4 for variable "x". In the case when partition_scaling_factor = 4, this function creates the new partition vector as follows: [0, 3, 3.5, 4, 4.5, 7, 9]

There are two options for this function,

* use_disc(default=m.discretization):: to regulate which is the base to add new partitions on
* use_solution(default=m.best_bound_sol):: to regulate which solution to use when adding new partitions on

This function can be accordingly modified by the user to change the behavior of the solver, and thus the convergence.

Alpine.amp_post_convexificationMethod
amp_post_convexification(m::Optimizer; use_disc = nothing)

Wrapper function to apply piecewise convexification of the problem for the bounding MIP model. This function talks to nonconvex_terms and convexification methods to finish the last step required during the construction of bounding model.

Alpine.bound_propagationMethod
detect_bound_from_aff(m::Optimizer)

Detect bounds from parse affine constraint. This function examines the one variable constraints such as x >= 5, x <= 5 or x == 5 and fetch the information to m.lvartight and m.uvartight.

Alpine.bound_tightening_wrapperMethod
bound_tightening_wrapper(m::Optimizer)

Entry point to the optimization-based bound-tightening (OBBT) algorithm. The aim of the OBBT algorithm is to sequentially tighten the variable bounds until a fixed point is reached.

Currently, two OBBT methods are implemented in optimization_based_bound_tightening.

* Bound-tightening with polyhedral relaxations (McCormick, Lambda for convex-hull)
* Bound-tightening with piecewise polyhedral relaxations: (with three partitions around the local feasible solution)

If no local feasible solution is obtained, the algorithm defaults to OBBT without partitions

Alpine.bounding_solveMethod

Alp.bounding_solve(m::Optimizer; kwargs...)

This step usually solves a convex MILP/MIQCP/MIQCQP problem for lower bounding the given minimization problem. It solves the problem built upon a piecewise convexification based on the discretization sictionary of some variables. See create_bounding_mip for more details of the problem solved here.

Alpine.build_constr_blockMethod

This utility function builds a constraint reference by repeating one operator with a vector variable references. For example, input => y, x[1,2,3], :+ output => (y = x[1] + x[2] + x[3])::Expr

Alpine.check_solution_historyMethod

checksolutionhistory(m::Optimizer, ind::Int)

Check if the solution is always the same within the last discconsecutiveforbid iterations. Return true if solution has stalled.

Alpine.correct_pointMethod
This function targets to address unexpected numerical issues when adding partitions in tight regions.
Alpine.create_bounding_mipMethod
create_bounding_mip(m::Optimizer; use_disc = nothing)

Set up a MILP bounding model base on variable domain partitioning information stored in use_disc. By default, if use_disc is not provided, it will use m.discretizations store in the Alpine model. The basic idea of this MILP bounding model is to use piecewise polyhedral/convex relaxations to tighten the basic relaxations of the original non-convex region. Among all presented partitions, the bounding model will choose one specific partition as the lower bound solution. The more partitions there are, the better or finer bounding model relax the original MINLP while the more efforts required to solve this MILP is required.

Alpine.detect_bilinear_termMethod
Future MONOMIAL Cluster
Recognize bilinear terms: x * y, where x and y are continous variables
Recognize multilinear terms: x1 * x2 * .. * xN, where all x_i ∀ i are continous variables
Recognize monomial terms: x^2 or x * x, where x is continuous
Alpine.detect_discretemulti_termMethod
Recognize prodcuts of binary variables and multilinear products

General Case : x1 * x2 .. * xN * y1 * y2 .. * yM
where x are binary variables, y are continous variables

Leads to BINLIN terms, with BINPROD, INTPROD, INTLIN if necessary
Alpine.detect_nonconvex_termsMethod
detect_nonconvex_terms(expr, m::Optimizer)

This function recognizes, stores, and replaces a sub-tree expr with available user-defined/built-in structures patterns. The procedure creates the required number of lifted variables based on the patterns that it is trying to recognize. Then, it goes through all built-in structures and performs operatins to convexify the problem.

Specific structure pattern information will be described formally.

Alpine.ebd_IMethod
This function is the same I() function described in Vielma and Nemhauser 2011.
Alpine.ebd_SMethod
This function is the same S() function described in Vielma and Nemhauser 2011.
Alpine.ebd_grayMethod
Built-in Encoding methods: gray encoding
This is a compatible encoding
Alpine.ebd_link_expressionMethod
This is a utility function used in ebd_link_xα() that constructing the mapping that links partitions with
combinations of binary variables.
Alpine.ebd_link_xαMethod
This is the function that translate the bounding constraints (α¹b⁰+α²b¹ <= x <= α¹b¹+α²b²)
with log # of binary variables, i.e., generate these constraints using log # of binary variables.
Alpine.ebd_σMethod
This function is the same σ() function described in Vielma and Nemhauser 2011.
Alpine.embedding_mapFunction
This function creates a mapping dictionary to link the right λ to the right bineary variables
based on how many partitions are required and a given encoding method.
Alpine.expr_arrangeargsMethod

Re-arrange children by their type. Only consider 3 types: coefficients(Int64, Float64), :ref, :calls Sequence arrangement is dependent on operator

Alpine.expr_flattenFunction

Recursively pre-process the expression by treating the coefficients TODO: this function requires a lot of refining. Most issues can be caused by this function.

Alpine.expr_isolate_constMethod

Converts ((a1*x[1])^2 + (a2x[2])^2 + ... + (a_nx[n])^2) to (a1^2*x[1]^2 + a2^2x[2]^2 + ... + a_n^2x[n]^2) Signs in the summation can be +/- Note: This function does not support terms of type (a*(x[1] + x[2]))^2 yet.

Alpine.expr_linear_to_affineMethod

This function takes a constraint/objective expression and converts it into a affine expression data structure Use the function to traverse linear expressions traverseexprlineartoaffine()

Alpine.expr_resolve_signFunction

This function is treats the sign in expression trees by cleaning the following structure: args ::+ || ::- ::Call || ::ref By separating the structure with some dummy treatments

Alpine.flatten_discretizationMethod
flatten_discretization(discretization::Dict)

Utility functions to eliminate all partition on discretizing variable and keep the loose bounds.

Alpine.get_candidate_disc_varsMethod

getcandidatedisc_vars(m:Optimizer)

A built-in method for selecting variables for discretization. This function selects all candidate variables in the nonlinear terms for discretization, if under a threshold value of the number of nonlinear terms.

Alpine.global_solveMethod

global_solve(m::Optimizer)

Perform global optimization algorithm that is based on the adaptive piecewise convexification. This iterative algorithm loops over bounding_solve and local_solve until the optimality gap between the lower bound (relaxed problem with min. objective) and the upper bound (feasible problem) is within the user prescribed limits. Each bounding_solve provides a lower bound that serves as the partitioning point for the next iteration (this feature can be modified given a different add_adaptive_partition). Each local_solve provides an incumbent feasible solution. The algorithm terminates when atleast one of these conditions are satisfied: time limit, optimality condition, or iteration limit.

Alpine.init_discMethod
init_disc(m::Optimizer)

This function initialize the dynamic discretization used for any bounding models. By default, it takes (.lvarorig, .uvarorig) as the base information. User is allowed to use alternative bounds for initializing the discretization dictionary. The output is a dictionary with MathProgBase variable indices keys attached to the :Optimizer.discretization.

Alpine.init_tight_boundMethod
init_tight_bound(m::Optimizer)

Initialize internal bound vectors (placeholders) to be used in other places. In this case, we don't have to mess with the original bound information.

Alpine.min_vertex_coverMethod

minvertexcover(m::Optimizer)

min_vertex_cover chooses the variables based on the minimum vertex cover algorithm for the interaction graph of nonlinear terms which are adaptively partitioned for global optimization. This option can be activated by setting disc_var_pick = 1.

Alpine.optimization_based_bound_tighteningMethod
optimization_based_bound_tightening(m:Optimizer; use_bound::Bool=true, use_tmc::Bool)

This function implements the OBBT algorithm to tighten the variable bounds. It utilizes either the basic polyhedral relaxations or the piecewise polyhedral relaxations (TMC) to tighten the bounds. The TMC has additional binary variables while performing OBBT.

The algorithm as two main parameters. The first is the use_tmc, which when set to true invokes the algorithm on the TMC relaxation. The second parameter use_bound takes in the objective value of the local solve solution stored in best_sol for performing OBBT. The use_bound option is set to true when the local solve is successful in obtaining a feasible solution, else this parameter is set to false.

For details, refer to section 3.1.1 of Nagarajan, Lu, Wang, Bent, Sundar, "An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs" link.

Several other user-input options can be used to tune the OBBT algorithm. For more details, see Presolve Options.

Alpine.pick_disc_varsMethod

pickdiscvars(m::Optimizer)

This function helps pick the variables for discretization. The method chosen depends on user-inputs. In case when indices::Int is provided, the method is chosen as built-in method. Currently, there are two built-in options for users as follows:

• max_cover (Alp.get_option(m, :disc_var_pick)=0, default): pick all variables involved in the non-linear term for discretization
• min_vertex_cover (Alp.get_option(m, :disc_var_pick)=1): pick a minimum vertex cover for variables involved in non-linear terms so that each non-linear term is at least convexified

For advanced usage, Alp.get_option(m, :disc_var_pick) allows ::Function inputs. User can provide his/her own function to choose the variables for discretization.

Alpine.post_objective_boundMethod
post_objective_bound(m::Optimizer, bound::Float64; kwargs...)

This function adds the upper/lower bounding constraint on the objective function for the optimization models solved within the OBBT algorithm.

Alpine.relaxation_bilinearMethod
relaxation_bilinear(
m::JuMP.Model,
z::JuMP.VariableRef,
x::JuMP.VariableRef,
y::JuMP.VariableRef,
lb_x::Number,
ub_x::Number,
lb_y::Number,
ub_y::Number)

General relaxation of a binlinear term (McCormick relaxation).

z >= JuMP.lower_bound(x)*y + JuMP.lower_bound(y)*x - JuMP.lower_bound(x)*JuMP.lower_bound(y)
z >= JuMP.upper_bound(x)*y + JuMP.upper_bound(y)*x - JuMP.upper_bound(x)*JuMP.upper_bound(y)
z <= JuMP.lower_bound(x)*y + JuMP.upper_bound(y)*x - JuMP.lower_bound(x)*JuMP.upper_bound(y)
z <= JuMP.upper_bound(x)*y + JuMP.lower_bound(y)*x - JuMP.upper_bound(x)*JuMP.lower_bound(y)
Alpine.relaxation_multilinear_binaryMethod
relaxation_multilinear_binary(
m::JuMP.Model,
z::JuMP.VariableRef,
x::Vector{VariableRef})

Applies Fortet linearization (see https://doi.org/10.1007/s10288-006-0015-3) for z = prod(x), where x is a vector of binary variables.

Alpine.resolve_var_boundsMethod
resolve_var_bounds(nonconvex_terms::Dict, discretization::Dict)

For discretization to be performed, we do not allow a variable being discretized to have infinite bounds.
The lifted/auxiliary variables may have infinite bounds and the function infers bounds on these variables. This process
can help speed up the subsequent solve times.

Only used in presolve bound tightening
Alpine.resolve_var_boundsMethod
resolve_var_bounds(m::Optimizer)

Resolve the bounds of the lifted variable using the information in lvartight and uvartight. This method only takes in known or trivial bounds information to deduce lifted variable bounds and to potentially avoid the cases of infinity bounds.

Alpine.set_mip_time_limitMethod
set_mip_time_limit(m::Optimizer)

An utility function used to dynamically regulate MILP solver time limits to fit Alpine solver's time limits.

Alpine.summary_statusMethod

This function summarizes the eventual solver status based on all available information recorded in the solver. The output status is self-defined which requires users to read our documentation to understand the details behind every status symbols.

Alpine.traverse_expr_linear_to_affineFunction

traverseexprlineartoaffine(expr, lhscoeffs=[], lhsvars=[], rhs=0.0, bufferVal=0.0, bufferVar=nothing, sign=1.0, level=0)

This function traverses the left hand side tree to collect affine terms. Updated status : possible to handle (x-(x+y(t-z))) cases where signs are handled properly

Alpine.update_disc_cont_varMethod
Ranking of variables involved in nonlinear terms for piecewise adaptive partitioning:
Ranked based on the absolute gap between each variable's solution from the lower-bounding MIP and the best feasible solution to the MINLP.
Currently doesn't support recursive convexification
Alpine.update_opt_gapMethod

updateoptgap(m::Optimizer)

Updates Alpine model's relative & absolute optimality gaps.

The relative gap calculation is

$$$\textbf{Gap} = \frac{|UB-LB|}{ϵ+|UB|}$$$

The absolute gap calculation is

|UB-LB|
Alpine.update_var_boundsMethod
update_var_bounds(m::Optimizer, discretization::Dict; len::Float64=length(keys(discretization)))

This function takes in a dictionary-based discretization information and convert them into two bounds vectors (lvar, uvar) by picking the smallest and largest numbers. User can specify a certain length that may contains variables that is out of the scope of discretization.

Output::

l_var::Vector{Float64}, u_var::Vector{Float64}