优化popt。jl
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脧锚脧赂`优化popt。jl`是一个包装包，集成https://github.com/jump-dev/Ipopt.jl[脧锚脧赂`Ipopt。jl`]与https://github.com/SciML/Optimization.jl[脧锚脧赂`优化。jl`]生态系统。 这允许您通过优化使用强大的Ipopt（内部点优化器）求解器。jl的统一接口。
Ipopt是用于大规模非线性优化的软件包，旨在找到形式的数学优化问题的（局部）解决方案:
哪里 是目标函数, 是约束函数和向量 和 表示约束的下限和上限，以及向量 和 是变量上的边界 .
方法
选项和参数
常用接口选项
以下选项可以作为关键字参数传递给 解决方案 并遵循共同的优化。jl接口:
* 最大的,最大的：最大迭代次数（复盖Ipopt的 最大值,最大值)
* 最大时间：以秒为单位的最长墙壁时间（复盖Ipopt的 max_wall_时间)
* 阿布斯托尔：绝对公差（ipopt不直接使用）
* [医]雷托尔：收敛容差（复盖Ipopt的 托尔)
* 详细,详细：控制输出详细程度（复盖Ipopt的 打印级别)
** 错误 或 0:无输出
** 真的 或 5:标准输出
**整数值0-12：不同的详细级别
Ipoptimizer构造函数选项
Ipopt特定的选项传递给 Ipoptimizer,ipoptimizer 构造函数。 最常用的选项可用作结构字段:
终止选项
* 可接受性_tol::Float64=1e-6：可接受的收敛公差（相对）
* 可接受_iter::Int=15：终止前可接受的迭代次数
* dual_inf_tol::Float64=1.0：双不可行性的期望阈值
* constr_viol_tol::Float64=1e-4：约束违反的期望阈值
* compl_inf_tol::Float64=1e-4：互补条件的期望阈值
线性求解器选项
* linear_solver::String="腮腺炎"：线性求解器使用
**默认值："腮腺炎"（包括Ipopt）
**HSL求解器："ma27"，"ma57"，"ma86"，"ma97"（需要https://github.com/jump-dev/Ipopt.jl?tab=readme-ov-file#linear-solvers[单独安装])
**其他:"pardiso","spral"(需要https://github.com/jump-dev/Ipopt.jl?tab=readme-ov-file#linear-solvers[单独安装])
* linear_system_scaling::String="无"：缩放线性系统的方法。 对HSL求解器使用"mc19"。
NLP缩放选项
* nlp_scaling_method::String="基于梯度的":Nlp的缩放方法
**选项："无"、"用户缩放"、"基于梯度"、"基于平衡"
* nlp_scaling_max_gradient::Float64=100.0：缩放后的最大梯度
屏障参数选项
* mu_strategy::String="单调"：屏障参数的更新策略（"单调"，"自适应"）
* mu_init::Float64=0.1:障碍参数的初始值
* mu_oracle::String="质量函数"：自适应mu策略的Oracle
Hessian选项
* hessian_approximation::String="精确"：如何近似Hessian
** "确切"：使用精确的Hessian
** "有限记忆"：使用L-BFGS近似
* limited_memory_max_history::Int=6:L-BFGS的历史尺寸
* limited_memory_update_type::String="bfgs"：准牛顿更新公式（"bfgs"，"sr1"）
附加选项字典
对于作为结构字段不可用的Ipopt选项，请使用 附加选项 字典:
opt = IpoptOptimizer(
linear_solver = "ma57",
additional_options = Dict(
"derivative_test" => "first-order",
"derivative_test_tol" => 1e-4,
"fixed_variable_treatment" => "make_parameter",
"alpha_for_y" => "primal"
)
)可用选项的完整列表记录在https://coin-or.github.io/Ipopt/OPTIONS.html[Ipopt选项参考]。
选项优先级
选项遵循此优先级顺序（从最高到最低）:
-
传递给的通用接口参数
解决方案（例如,[医]雷托尔,最大的,最大的) -
选项
附加选项字典 -
结构字段值
Ipoptimizer,ipoptimizer
具有多个选项源的示例:
opt = IpoptOptimizer(
acceptable_tol = 1e-6, # Struct field
mu_strategy = "adaptive", # Struct field
linear_solver = "ma57", # Struct field (needs HSL)
print_timing_statistics = "yes", # Struct field
additional_options = Dict(
"alpha_for_y" => "primal", # Not a struct field
"max_iter" => 500 # Will be overridden by maxiters below
)
)
sol = solve(prob, opt;
maxiters = 1000, # Overrides max_iter in additional_options
reltol = 1e-8 # Sets Ipopt's tol
)例子：
基本无约束优化
Rosenbrock函数可以使用 Ipoptimizer,ipoptimizer:
using Optimization, OptimizationIpopt
using Zygote
rosenbrock(x, p) = (p[1] - x[1])^2 + p[2] * (x[2] - x[1]^2)^2
x0 = zeros(2)
p = [1.0, 100.0]
# Ipopt requires gradient information
optfunc = OptimizationFunction(rosenbrock, AutoZygote())
prob = OptimizationProblem(optfunc, x0, p)
sol = solve(prob, IpoptOptimizer())retcode: Success
u: 2-element Vector{Float64}:
0.9999999999999899
0.9999999999999792
盒约束优化
添加框约束以限制搜索空间:
using Optimization, OptimizationIpopt
using Zygote
rosenbrock(x, p) = (p[1] - x[1])^2 + p[2] * (x[2] - x[1]^2)^2
x0 = zeros(2)
p = [1.0, 100.0]
optfunc = OptimizationFunction(rosenbrock, AutoZygote())
prob = OptimizationProblem(optfunc, x0, p;
lb = [-1.0, -1.0],
ub = [1.5, 1.5])
sol = solve(prob, IpoptOptimizer())retcode: Success
u: 2-element Vector{Float64}:
0.9999999942690103
0.9999999885017182
非线性约束优化
用非线性相等和不等式约束求解问题:
using Optimization, OptimizationIpopt
using Zygote
# Objective: minimize x[1]^2 + x[2]^2
objective(x, p) = x[1]^2 + x[2]^2
# Constraint: x[1]^2 + x[2]^2 - 2*x[1] = 0 (equality)
# and x[1] + x[2] >= 1 (inequality)
function constraints(res, x, p)
res[1] = x[1]^2 + x[2]^2 - 2*x[1] # equality constraint
res[2] = x[1] + x[2] # inequality constraint
end
x0 = [0.5, 0.5]
optfunc = OptimizationFunction(objective, AutoZygote(); cons = constraints)
# First constraint is equality (lcons = ucons = 0)
# Second constraint is inequality (lcons = 1, ucons = Inf)
prob = OptimizationProblem(optfunc, x0;
lcons = [0.0, 1.0],
ucons = [0.0, Inf])
sol = solve(prob, IpoptOptimizer())retcode: Success
u: 2-element Vector{Float64}:
0.29289321506718563
0.7071067774417749
使用有限内存BFGS近似
对于计算确切的Hessian是昂贵的大规模问题:
using Optimization, OptimizationIpopt
using Zygote
# Large-scale problem
n = 100
rosenbrock_nd(x, p) = sum(p[2] * (x[i+1] - x[i]^2)^2 + (p[1] - x[i])^2 for i in 1:n-1)
x0 = zeros(n)
p = [1.0, 100.0]
# Using automatic differentiation for gradients only
optfunc = OptimizationFunction(rosenbrock_nd, AutoZygote())
prob = OptimizationProblem(optfunc, x0, p)
# Use L-BFGS approximation for Hessian
sol = solve(prob, IpoptOptimizer(
hessian_approximation = "limited-memory",
limited_memory_max_history = 10);
maxiters = 1000)retcode: Success
u: 100-element Vector{Float64}:
1.000000000045498
0.9999999999674329
0.9999999999760841
1.0000000000439668
0.9999999999656436
0.9999999999982687
1.0000000000174465
0.9999999999961191
0.9999999999984535
1.0000000000270632
⋮
1.0000000000791858
0.9999999998741496
0.9999999998274194
0.9999999997875975
0.9999999993748642
0.9999999990813158
0.9999999981939615
0.9999999966259885
0.9999999933639284
投资组合优化示例
一个具有约束的投资组合优化的实际例子:
using Optimization, OptimizationIpopt
using Zygote
using LinearAlgebra
# Portfolio optimization: minimize risk subject to return constraint
n_assets = 5
μ = [0.05, 0.10, 0.15, 0.08, 0.12] # Expected returns
Σ = [0.05 0.01 0.02 0.01 0.00; # Covariance matrix
0.01 0.10 0.03 0.02 0.01;
0.02 0.03 0.15 0.02 0.03;
0.01 0.02 0.02 0.08 0.02;
0.00 0.01 0.03 0.02 0.06]
target_return = 0.10
# Objective: minimize portfolio variance
portfolio_risk(w, p) = dot(w, Σ * w)
# Constraints: sum of weights = 1, expected return >= target
function portfolio_constraints(res, w, p)
res[1] = sum(w) - 1.0 # Sum to 1 (equality)
res[2] = dot(μ, w) - target_return # Minimum return (inequality)
end
optfunc = OptimizationFunction(portfolio_risk, AutoZygote();
cons = portfolio_constraints)
w0 = fill(1.0/n_assets, n_assets)
prob = OptimizationProblem(optfunc, w0;
lb = zeros(n_assets), # No short selling
ub = ones(n_assets), # No single asset > 100%
lcons = [0.0, 0.0], # Equality and inequality constraints
ucons = [0.0, Inf])
sol = solve(prob, IpoptOptimizer();
reltol = 1e-8,
verbose = 5)
println（"最佳权重："，sol。u)
println("期望返回:",dot(μ,sol.u))
println（"Portfolio variance："，sol。目的)This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 5
Number of nonzeros in inequality constraint Jacobian.: 5
Number of nonzeros in Lagrangian Hessian.............: 15
Total number of variables............................: 5
variables with only lower bounds: 0
variables with lower and upper bounds: 5
variables with only upper bounds: 0
Total number of equality constraints.................: 1
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 1
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 3.1200000e-02 0.00e+00 3.43e-02 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.2177561e-02 0.00e+00 1.30e-02 -1.7 1.78e-02 - 9.93e-01 1.00e+00h 1
2 3.0759152e-02 0.00e+00 4.26e-02 -2.5 5.87e-02 - 9.79e-01 1.00e+00f 1
3 2.9846484e-02 0.00e+00 2.83e-08 -2.5 1.07e-01 - 1.00e+00 1.00e+00f 1
4 2.8068833e-02 0.00e+00 2.47e-02 -3.8 3.60e-02 - 8.65e-01 1.00e+00f 1
5 2.7401391e-02 0.00e+00 1.50e-09 -3.8 2.41e-02 - 1.00e+00 1.00e+00f 1
6 2.7234938e-02 0.00e+00 1.46e-04 -5.7 1.03e-02 - 9.79e-01 1.00e+00f 1
7 2.7232343e-02 0.00e+00 1.84e-11 -5.7 1.85e-03 - 1.00e+00 1.00e+00f 1
8 2.7230500e-02 2.22e-16 2.51e-14 -8.6 1.40e-04 - 1.00e+00 1.00e+00f 1
Number of Iterations....: 8
(scaled) (unscaled)
Objective...............: 2.7230500043271134e-02 2.7230500043271134e-02
Dual infeasibility......: 2.5091040356528538e-14 2.5091040356528538e-14
Constraint violation....: 2.2204460492503131e-16 2.2204460492503131e-16
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 6.4320485361915960e-09 6.4320485361915960e-09
Overall NLP error.......: 6.4320485361915960e-09 6.4320485361915960e-09
Number of objective function evaluations = 9
Number of objective gradient evaluations = 9
Number of equality constraint evaluations = 9
Number of inequality constraint evaluations = 9
Number of equality constraint Jacobian evaluations = 9
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations = 8
Total seconds in IPOPT = 1.938
EXIT: Optimal Solution Found.
Optimal weights: [0.23290714113956607, 0.16055161296302972, 0.0967086319310212, 0.08466825825418363, 0.4251643557121995]
Expected return: 0.09999999648873309
Portfolio variance: 0.027230500043271134
提示和最佳实践
-
*缩放*：当变量和约束被很好地缩放时，Ipopt表现更好。 如果变量具有非常不同的幅度，请考虑规范化您的问题。
-
*初始点*：尽可能提供良好的初始猜测。 Ipopt是一个本地优化器，解决方案质量取决于起点。
-
*Hessian近似*：对于大问题或Hessian计算昂贵时，使用
hessian_approximation="有限内存"在Ipoptimizer,ipoptimizer构造函数。 -
*线性求解器选择*：线性求解器的选择会显着影响性能。 对于大问题，请考虑使用HSL求解器（ma27、ma57、ma86、ma97）。 请注意，HSL求解器需要https://github.com/jump-dev/Ipopt.jl?tab=readme-ov-file#linear-solvers[单独安装]-请参阅Ipopt。安装说明的jl文档. 默认的流行性腮腺炎解算器适用于中小问题。
-
*约束公式*：Ipopt很好地处理相等约束。 在可能的情况下，将约束制定为相等而不是成对的不等式。
-
*温启动*：在解决一系列类似问题时，使用前一个问题的解决方案作为下一个问题的初始点。 您可以从以下开始启用warm
Ipoptimizer(warm_start_init_point="是").