shgo

Simplicial Homology Global Optimisation

shgo - Stefan Endres
shgo | Stefan Endres

Corresponding author for shgo: Stefan Endres

Introduction

The simplicial homology global optimisation (shgo) algorithm is a promising, recently published global optimisation (GO) algorithm [1]. The software implementation of the algorithm has been shown to be highly competitive when compared to state of the art commercial and open-source optimisation software. It was shown that shgo had the highest performance on a test suite of constrained problems [1]. The shgo algorithm is relatively unique among GO algorithms in that it also returns all other local and global minimum in addition to the global minimum (using novel, rigorously proven methods that detect the homological properties of the objective function surface). Therefore, shgo is highly appropriate for problems where the local minima are desired, such as energy surface problems or problems with more than one global solution.

As a general purpose solver, shgo solves any general class of optimisation problem. However, it is most appropriate for solving a global and derivative free optimisation problem. It is simple to use, requiring only a black-box (or “oracle”) function input. Therefore it is especially appropriate for science and engineering problems that have embedded simulations or has a highly complex model structure.

By its nature black-box problems are difficult to solve to global optimality, therefore it is unlikely that a problem with more than 10 variables (dimensions) can be solved quickly with the minimum input of the objective function. If you have more information about your model structure (such as known bounds, symmetry or gradients) it is generally recommended to provide shgo with this information which will greatly speed up performance and allow you to solve problems with more variables.

The remainder of this introductory section describes the theory behind shgo and the current features of the software. If you want to quickly learn how to implement shgo on a problem of your own, you can skip to the Installation section and the Non-linear constraints example, which demonstates how to program the most general case of a global optimisation problem.

Global and Derivative Free Optimisation

The shgo algorithm is appropriate for solving general purpose NLP and blackbox optimisation problems to global optimality (low dimensional problems) [1-2]. This class of optimisation is also know as CDFO (constrained derivative free optimisation). In general, the optimisation problems are of the form::

\begin{eqnarray} \nonumber \min_x && f(x), x \in \mathbb{R}^n \\\ \nonumber \text{s.t.} && g_i(x) \ge 0, ~ \forall i = 1,…,m \\\ \nonumber && h_j(x) = 0, ~\forall j = 1,…,p \end{eqnarray}

where $x$ is a vector of one or more variables. $f(x)$ is the objective function $f: \mathbb{R}^n \rightarrow \mathbb{R}$

$g_i(x)$ are the inequality constraints $\mathbb{g}: \mathbb{R}^n \rightarrow \mathbb{R}^m$

$h_j(x)$ are the equality constraints $\mathbb{h}: \mathbb{R}^n \rightarrow \mathbb{R}^p$

Optionally, the lower and upper bounds $x_l \le x \le x_u$ for each element in $x$ can also be specified using the bounds argument.

While most of the theoretical advantages of shgo are only proven for when $f(x)$ is a Lipschitz smooth function. The algorithm is also proven to converge to the global optimum for the more general case where $f(x)$ is non-continuous, non-convex and non-smooth iff the default sampling method is used [2].

Simplicial Homology Global Optimisation Theory

In order to understand the properties of shgo some background theory is required. An important facet of shgo} is the concept of homology group growth which can be used by an optimisation practitioner as a visual guide of the number of local and global solutions to a problem of arbitrarily high dimensions. In addition a measure of the mutli-modality and the geometric sparsity of solutions of the optimisation problem instance can be deduced.

In brief the algorithm utilises concepts from combinatorial integral homology theory to find sub-domains which are, approximately, locally convex and provides characterisations of the objective function as the algorithm progresses. For example on the following objective function surface:

The $k-$chain $C(\mathcal{H}^k), k = n + 1$ of simplices in $\textrm{st}\left( v_1 \right)$ forms a boundary cycle $\partial(C(\mathcal{H}^{n + 1})) = \overline{v_{11} v_{9}} - \overline{v_{9} v_{14}} + \overline{v_{14} v_{4}} - \overline{v_{4} v_{11}}$ with $\partial\left(\partial(C(\mathcal{H}^{n + 1}))\right) = \emptyset$

The starting point $v_1$ and its corresponding constraints are then passed to local-minimisation routines during each iteration. The algorithm provides many additional guarantees such always passing active constraints (in $g_i(x)$) when a local minimum lies on it. For example, here the star domain around vertex $v_7$ is bounded by the global constraints of the problem:

Another important property of shgo is that it is proven that one and only one starting point for every strictly unique minimum is produced. So the number of locally convex subdomains do not increase with increased sampling of the objective function, rather only the constraints of the convex subdomains are refined:

This distinguishes it from many other global optimisation algorithms using graph theory and clustering methods that often show poor performance by producing starting points that converge to the same local minimum [1-2]. These guarantees are proven to hold for Lipschitz smooth functions of arbitrarily high dimensions. In addition since the locally convex sub-domains are processed, shgo can concentrate on the global search. This circumvents the need to specify the usual trade-off between a local and global search.

This is accomplished in several steps. First the construction of a simplicial complex $\mathcal{H}$ built up from the sampling points mapped through $f$ as vertices following the constructions described in \cite{Endres2018}. Next a homomorphism is found between $\mathcal{H}$ and $\mathcal{K}$; another simplicial complex which exists on an abstract constructed surface $\mathcal{S}$. The $n$-dimensional manifold $\mathcal{S}$ is a connected g sum of g tori $S := S_0\,#\,S_1\,#\,\cdots\,#\,S_{g - 1}$. The figures below demonstrate this construction geometrically in the 2-dimensional case. By using an extension of Brouwer’s fixed point theorem (Henle, 1979) adapted to handle non-linear constraints, it is proven that each of the “minimiser points” in the figure below corresponds to a sub-domain containing a unique local-minima when the problem is adequately sampled. Through the Invariance Theorem [17] and the Eilenberg-Steenrod Axioms [17-18], we draw another homomorphism between the surfaces of $f$ and $\mathcal{S}$.

The process of puncturing a hypersphere at a minimiser point in a compact search space. Start by identifying a minimiser point in the $\mathcal{H}^1$ ($\cong~\mathcal{K}^1$) graph. By construction, our initial complex exists on the (hyper-)surface of an $n$-dimensional torus $\mathcal{S}_0$ such that the rest of $\mathcal{K}^1$ is connected and compact. We puncture a hypersphere at the minimiser point and identify the resulting edges (or ($n-1$)-simplices in higher dimensional problems). Next we shrink (a topoligical (ie continuous) transformation) the remainder of the simplicial complex to the faces and vertices of our (hyper-)plane model. Make the appropriate identifications for $\mathcal{S}_0$ and glue the identified and connected face $z$ (a ($n-1$)-simplex) that resulted from the hypersphere puncture. The other faces (ie ($n-1$)-simplices) are connected in the usual way for tori constructions.

The process of puncturing a new hypersphere on $\mathcal{S}{0}\,#\,\mathcal{S}{1}$ can be repeated for any new minimiser point without loss of generality producing $S := S_0\,#\,S_1\,#\,\cdots\,#\,S_{g - 1} \qquad (g\text{ times})$.

Visual demonstration on surfaces with non-linear constraints, the shaded region is unfeasible. The vertices of the points mapped to infinity have undirected edges, therefore they do not form simplicial complexes in the integral homology. The surfaces of each disconnected simplicial complex $\mathcal{K}_i$ can be constructed from the compact version of the invariance theorem. The rank of the abelian homology groups $\mathbf{H}_1(\mathcal{K}_i)$ is additive over arbitrary direct sums.

The full outline of the algorithm can be downloaded here. Detailed description of the properties and their proofs can be found in [1-2].

The local search method may be specified using the minimizer_kwargs parameter which is inputted to scipy.optimize.minimize. By default the SLSQP method is used. Other local minimisation methods more suited to the problem can also be used. In general it is recommended to use the SLSQP or COBYLA local minimization if inequality constraints are defined for the problem since the other methods do not use constraints.

The sobol method points are generated using the Sobol [3] sequence. The primitive polynomials and various sets of initial direction numbers for generating Sobol sequences is provided in [4] by Frances Kuo and Stephen Joe. The original program sobol.cc (MIT) is available and described at http://web.maths.unsw.edu.au/~fkuo/sobol/ translated to Python 3 by Carl Sandrock 2016-03-31.

The deterministic sampling and refinement of shgo provides a more robust alternative to other algorithms commonly used in energy optimisation which rely on random sampling such as monte carlo methods [5] and bashinhopping [6] which can have varying performance.

The algorithm is generally applicable to low dimensional black problems (~10-dimensional problems) unless more information can be supplied to the algorithm. This is not necessarily only gradients and hessians. For example if it is known that the decision variables of the objective function are symmetric, then the symmetry option can be used in order to solve problems with hundreds of variables.

Summary of shgo features

• Convergence to a global minimum assured.
• Allows for non-linear constraint in the problem statement.
• Extracts all the minima in the limit of an adequately sampled search space (ie attempts to find all the (quasi-)equilibrium solutions).
• Progress can be tracked after every iteration through the calculated homology groups.
• Competitive performance compared to state of the art black-box solvers.
• All of the above properties hold for non-continuous functions with non-linear constraints assuming the search space contains any sub-spaces that are continuous and convex.

Performance summary

Open-source black-box algorithms

The shgo algorithm only makes use of function evaluations without requiring the derivatives of objective functions. This makes it applicable to black-box global optimisation problems. Here we compare the SHGO and TGO algorithms with the SciPy implementation of basinhopping (BH) [6] and differential evolution (DE) orignally proposed Storn and Price [7]. These algorithms were chosen because the open source versions are readily available in the SciPy project. The test suite contains multi-modal problems with box constraints, they are described in detail in infinity77.net/global_optimization/. We used the stopping criteria pe = 0.01% for shgo and tgo. Any local function evaluations were added to the global count. For the stochastic algorithms (BH and DE) the starting points provided by the test suite were used. For every test the algorithm was terminated if the global minimum was not found after 10 minutes of processing time and the test was flagged as a fail.

This figure shows the performance profiles for SHGO, TGO, DE and BH on the SciPy benchmarking test suite using function evaluations and processing run time as performance criteria:

Performance profiles zoomed in to the range of f.e. = [0, 1000] function evaluations and [0, 0.4] seconds run time:

From the figures it can be observed that for this problem set shgo-sobol was the best performing algorithm, followed closely by tgo and shgo-simpl. The zoomed figure provides a clearer comparison between these three algorithms. While the performance of all 3 algorithms are comparable, shgo-Sobol tends to outperform shgo, solving more problems or a given number of function evaluations. This is expected since, for the same sampling point sequence, tgo produced more than one starting point in the same locally convex domain while shgo is guaranteed to only produce one after adequate sampling. While shgo-simpl has the advantage of having the theoretical guarantee of convergence, the sampling sequence has not been optimised yet requiring more function evaluations with every iteration than shgo-sobol.

Recently published black-box algorithms

A recent review and experimental comparison of 22 derivative-free optimisation algorithms by Rios and Sahinidis [8] concluded that global optimisation solvers solvers such as TOMLAB/MULTI-MIN, TOMLAB/GLCCLUSTER, MCS and TOMLAB/LGO perform better, on average, than other derivative-free solvers in terms of solution quality within 2500 function evaluations. Both the TOMLAB/GLC-CLUSTER and MCS Huyer and Neumaier (1999) implementations are based on the well-known DIRECT (DIviding RECTangle) algorithm [9].

The DISIMPL (DIviding SIMPLices) algorithm was recently proposed by Paulavičius and Žilinskas [11-13]. The experimental investigation in [11] shows that the proposed simplicial algorithm gives very competitive results compared to the DIRECT algorithm [9]. DISIMPL has been extended in [10-11]. The Gb-DISIMPL (Globally-biased DISIMPL) was compared in Paulavičius et al. (2014) [11] to the DIRECT and DIRECT-l methods in extensive numerical experiments on 800 multidimensional multiextremal. Gb-DISIMPL was shown to provide highly competative results compared the other algorithms.

More recently the Lc-DISIMPL variant of the algorithm was developed to handle optimisation problems with linear constraints [13]. Below we use an extract of the results with the highest performing Lc-DISIMPL algorithm (Lc-DISIMPL-v) and DIRECT-L1 with the best performaning parameters (pp = 10). The full table can be found at here. From the table it can be seen shgo provides competative results compared to the other algorithms:

Algorithm:shgo-simplshgo-sobLc-DISIMPL-vPSwarm (avg)LGODIRECT-L1
horst-1972471329$^{b(3)}$2457287$^a$
horst-2101154241645265$^a$
horst-36754436495$^a$
horst-4102581143958293$^a$
horst-520158134377$^a$
horst-6225910110847611$^a$
horst-710151038052177$^a$
hs02124231891891397
hs02424153118180919$^a$
hs03537416303161885>100000
hs036105208396275625$^a$
hs0377263186160105167$^a$
hs03822510293379585762217401
hs0441993520186$^{b(9)}$3246490283
hs076563754820322119135
s22416616549121247$^a$
s23199992137236619961261
s23224153119182619$^a$
s2501052083673225$^a$
s2517263186129105757$^a$
bunnag1344763021418841529
bunnag246361625276454>100000

Average668836630116841>17213

$a$ result is outside the feasible region

$b(t)$ $t$ out of 10 times the global solution was not reached

Lc-DISIMPL-v, PSwarm (avg), DIRECT-L1 results produced by Paulavičius & Žilinskas (2016)

Performance profiles for shgo, TGO, Lc-DISIMPL, LGO, PSwarm and DIRECT- L1 algorithms on linearly constrained test problems. The figure displays the fraction test suite problems that can be solved within a given number of objective function evaluations. The results for Lc-DISIMPL-v, PSwarm (avg), DIRECT-L1 were produced by

LGO (Lipschitz-continuous Global Optimizer) [14]

J. D. Pintér, Nonlinear optimization with gams /lgo, J. of Global Opti- mization 38 (1) (2007) 79–101. doi:10.1007/s10898-006-9084-2. URL http://dx.doi.org/10.1007/s10898-006-9084-2

It can be seen that shgo with the simplicial and Sobol sampling method generally outperforms every other algorithm. The only exception is the better early performance by Lc-DISIMPL. This is attributed to Lc-DISIMPL’s initiation step solving the set of equations in the linear constraints. In the test problems where the global minimum lie on a vertex of this convex hull, the algorithm immediately terminates without a global sampling phase. For more gen- eral, non-linear constraints it would not be possible to use this feature of Lc-DISIMPL.

Installation

Stable:

$pip install shgo  Latest: $ git clone https://bitbucket.org/upiamcompthermo/shgo
$cd shgo$ python setup.py install
$python setup.py test  Examples Unimodal function: Rosenbrock Bounded variables First consider the problem of minimizing the Rosenbrock function which is unimodal in 2-dimensions. This function is implemented in rosen in scipy.optimize >>> from scipy.optimize import rosen >>> from shgo import shgo >>> bounds = [(0,2), (0, 2)] >>> result = shgo(rosen, bounds) >>> result.x, result.fun (array([ 1., 1.]), 3.6584112734652932e-19)  Unbounded variables Note that bounds determine the dimensionality of the objective function and is therefore a required input, however you can specify empty bounds using None or objects like numpy.inf which will be converted to large float numbers. >>> bounds = [(None, None), ]*2 >>> result = shgo(rosen, bounds) >>> result.x array([ 0.99999555, 0.99999111])  Multimodal function: Eggholder Mapping local minima Next we consider the Eggholder function, a problem with several local minima and one global minimum. We will demonstrate the use of some of the arguments and capabilities of shgo. >>> from shgo import shgo >>> import numpy as np >>> def eggholder(x): ... return (-(x[1] + 47.0) ... * np.sin(np.sqrt(abs(x[0]/2.0 + (x[1] + 47.0)))) ... - x[0] * np.sin(np.sqrt(abs(x[0] - (x[1] + 47.0)))) ... ) ... >>> bounds = [(-512, 512), (-512, 512)]  shgo has two built-in low discrepancy sampling sequences. The default simplicial and the sobol sequence. First we will input 30 initial sampling points of the Sobol sequence >>> result = shgo(eggholder, bounds, n=30, sampling_method='sobol') >>> result.x, result.fun (array([ 512. , 404.23180542]), -959.64066272085051)  shgo also has a return for any other local minima that was found, these can be called using: >>> result.xl, result.funl (array([[ 512. , 404.23180542], [ 283.07593402, -487.12566542], [-294.66820039, -462.01964031], [-105.87688985, 423.15324143], [-242.97923629, 274.38032063], [-506.25823477, 6.3131022 ], [-408.71981195, -156.10117154], [ 150.23210485, 301.31378508], [ 91.00922754, -391.28375925], [ 202.8966344 , -269.38042147], [ 361.66625957, -106.96490692], [-219.40615102, -244.06022436], [ 151.59603137, -100.61082677]]), array([-959.64066272, -718.16745962, -704.80659592, -565.99778097, -559.78685655, -557.36868733, -507.87385942, -493.9605115 , -426.48799655, -421.15571437, -419.31194957, -410.98477763, -202.53912972]))  These results are useful in applications where there are many global minima and the values of other global minima are desired or where the local minima can provide insight into the system such as for example morphologies in physical chemistry [15]. Improving results Now suppose we want to find a larger number of local minima (or we hope to find a lower minimum than the current best). This can be accomplished for example by increasing the amount of sampling points or the number of iterations. We’ll increase the number of sampling points to 60 and the number of iterations to 3 increased from the default 100 for a total of 60 x 3 = 180 initial sampling points. >>> result_2 = shgo(eggholder, bounds, n=60, iters=5, sampling_method='sobol') >>> len(result.xl), len(result_2.xl) (13, 39)  Note that there is a difference between specifying arguments for ex. n=180, iters=1 and n=60, iters=3. In the first case the promising points contained in the minimiser pool is processed only once. In the latter case it is processed every 60 sampling points for a total of 3 iterations. Non-linear constraints: cattle feed problem (HS73) To demonstrate solving problems with non-linear constraints consider the following example from Hock and Schittkowski problem 73 (cattle-feed) [16]:: \begin{eqnarray} \nonumber \textrm{minimize}: f(x) =&& 24.55 x_1 + 26.75 x_2 + 39 x_3 + 40.50 x_4 & \\\ \nonumber \text{s.t.} && 2.3 x_1 + 5.6 x_2 + 11.1 x_3 + 1.3 x_4 - 5 &\ge 0, \\\ \nonumber && 12 x_1 + 11.9 x_2 + 41.8 x_3 + 52.1 x_4 - 21 & \\\ \nonumber && -1.645 \sqrt{0.28 x_1^2 + 0.19 x_2^2 + 20.5 x_3^2 + 0.62 x_4^2} &\ge 0, \\\ \nonumber && x_1 + x_2 + x_3 + x_4 - 1 &= 0, \\\ \nonumber && 0 \le x_i \le 1 ~~ \forall i \end{eqnarray} Approx. answer [4]:$f([0.6355216, -0.12e^{-11}, 0.3127019, 0.05177655]) = 29.894378$ >>> from shgo import shgo >>> import numpy as np >>> def f(x): # (cattle-feed) ... return 24.55*x[0] + 26.75*x[1] + 39*x[2] + 40.50*x[3] ... >>> def g1(x): ... return 2.3*x[0] + 5.6*x[1] + 11.1*x[2] + 1.3*x[3] - 5 # >=0 ... >>> def g2(x): ... return (12*x[0] + 11.9*x[1] +41.8*x[2] + 52.1*x[3] - 21 ... - 1.645 * np.sqrt(0.28*x[0]**2 + 0.19*x[1]**2 ... + 20.5*x[2]**2 + 0.62*x[3]**2) ... ) # >=0 ... >>> def h1(x): ... return x[0] + x[1] + x[2] + x[3] - 1 # == 0 ... >>> cons = ({'type': 'ineq', 'fun': g1}, ... {'type': 'ineq', 'fun': g2}, ... {'type': 'eq', 'fun': h1}) >>> bounds = [(0, 1.0),]*4 >>> res = shgo(f, bounds, iters=3, constraints=cons) >>> res fun: 29.894378159142136 funl: array([ 29.89437816]) message: 'Optimization terminated successfully.' nfev: 119 nit: 3 nlfev: 40 nljev: 0 success: True x: array([ 6.35521569e-01, 1.13700270e-13, 3.12701881e-01, 5.17765506e-02]) xl: array([[ 6.35521569e-01, 1.13700270e-13, 3.12701881e-01, 5.17765506e-02]]) >>> g1(res.x), g2(res.x), h1(res.x) (-5.0626169922907138e-14, -2.9594104944408173e-12, 0.0)  Advanced features (Under construction) Model structure and performance Stopping criteria Parallelization Code parameters func : callable  The objective function to be minimized. Must be in the form f(x, *args), where x is the argument in the form of a 1-D array and args is a tuple of any additional fixed parameters needed to completely specify the function. bounds : sequence  Bounds for variables. (min, max) pairs for each element in x, defining the lower and upper bounds for the optimizing argument of func. It is required to have len(bounds) == len(x). len(bounds) is used to determine the number of parameters in x. Use None for one of min or max when there is no bound in that direction. By default bounds are (None, None). args : tuple, optional  Any additional fixed parameters needed to completely specify the objective function. constraints : dict or sequence of dict, optional  Constraints definition. Function(s)$\mathbb{R}^n$in the form:$g(x) \le 0$applied as$\mathbb{g}: \mathbb{R}^n \rightarrow \mathbb{R}^mh(x) = 0$applied as$\mathbb{g}: \mathbb{R}^n \rightarrow \mathbb{R}^p$Each constraint is defined in a dictionary with fields: * type : str Constraint type: 'eq' for equality$h(x), 'ineq' for inequality \$g(x).
* fun : callable
The function defining the constraint.
* jac : callable, optional
The Jacobian of fun (only for SLSQP).
* args : sequence, optional
Extra arguments to be passed to the function and Jacobian.


Equality constraint means that the constraint function result is to be zero whereas inequality means that it is to be non-negative. Note that COBYLA only supports inequality constraints.

NOTE: Only the COBYLA and SLSQP local minimize methods currently support constraint arguments. If the constraints sequence used in the local optimization problem is not defined in minimizer_kwargs and a constrained method is used then the global constraints will be used. (Defining a constraints sequence in minimizer_kwargs means that constraints will not be added so if equality constraints and so forth need to be added then the inequality functions in constraints need to be added to minimizer_kwargs too).

n : int, optional


Number of sampling points used in the construction of the simplicial complex. Note that this argument is only used for sobol and other arbitrary sampling_methods.

iters : int, optional


Number of iterations used in the construction of the simplicial complex.

callback : callable, optional


Called after each iteration, as callback(xk), where xk is the current parameter vector.

minimizer_kwargs : dict, optional


Extra keyword arguments to be passed to the minimizer scipy.optimize.minimize Some important options could be:

* method : str
The minimization method (e.g. SLSQP)
* args : tuple
Extra arguments passed to the objective function (func) and
its derivatives (Jacobian, Hessian).
* options : dict, optional
Note that by default the tolerance is specified as {ftol: 1e-12}


options : dict, optional


A dictionary of solver options. Many of the options specified for the global routine are also passed to the scipy.optimize.minimize routine. The options that are also passed to the local routine are marked with an (L)

Stopping criteria, the algorithm will terminate if any of the specified criteria are met. However, the default algorithm does not require any to be specified:

* maxfev : int (L)
Maximum number of function evaluations in the feasible domain.
(Note only methods that support this option will terminate
the routine at precisely exact specified value. Otherwise the
criterion will only terminate during a global iteration)
* f_min
Specify the minimum objective function value, if it is known.
* f_tol : float
Precision goal for the value of f in the stopping
criterion. Note that the global routine will also
terminate if a sampling point in the global routine is
within this tolerance.
* maxiter : int
Maximum number of iterations to perform.
* maxev : int
Maximum number of sampling evaluations to perform (includes
searching in infeasible points).
* maxtime : float
Maximum processing runtime allowed
* minhgrd : int
Minimum  homology group rank differential. The homology group of the
objective function is calculated (approximately) during every
iteration. The rank of this group has a one-to-one correspondence
with the number of locally convex subdomains in the objective
function (after adequate sampling points each of these subdomains
contain a unique global minima). If the difference in the hgr is 0
between iterations for maxhgrd specified iterations the
algorithm will terminate.


Objective function knowledge:

* symmetry : bool
Specify True if the objective function contains symmetric variables.
The search space (and therefore performance) is decreased by O(n!).

* jac : bool or callable, optional
Jacobian (gradient) of objective function. Only for CG, BFGS,
Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg. If jac is a
Boolean and is True, fun is assumed to return the gradient along
with the objective function. If False, the gradient will be
estimated numerically. jac can also be a callable returning the
gradient of the objective. In this case, it must accept the same
arguments as fun. (Passed to scipy.optimize.minmize automatically)

* hess, hessp : callable, optional
Hessian (matrix of second-order derivatives) of objective function
or Hessian of objective function times an arbitrary vector p.
Only for Newton-CG, dogleg, trust-ncg. Only one of hessp or hess
needs to be given. If hess is provided, then hessp will be ignored.
If neither hess nor hessp is provided, then the Hessian product
will be approximated using finite differences on jac. hessp must
compute the Hessian times an arbitrary vector.
(Passed to scipy.optimize.minmize automatically)


Algorithm settings:

* minimize_every_iter : bool
If True then promising global sampling points will be passed to a
local minimisation routine every iteration. If False then only the
final minimiser pool will be run. Defaults to False.
* local_iter : int
Only evaluate a few of the best minimiser pool candiates every
iteration. If False all potential points are passed to the local
minimsation routine.
* infty_constraints: bool
If True then any sampling points generated which are outside will
the feasible domain will be saved and given an objective function
value of numpy.inf. If False then these points will be discarded.
Using this functionality could lead to higher performance with
respect to function evaluations before the global minimum is found,
specifying False will use less memory at the cost of a slight
decrease in performance.


Feedback:

* disp : bool (L)
Set to True to print convergence messages.


sampling_method : str or function, optional


Current built in sampling method options are sobol and simplicial. The default simplicial uses less memory and provides the theoretical guarantee of convergence to the global minimum in finite time. The sobol method is faster in terms of sampling point generation at the cost of higher memory resources and the loss of guaranteed convergence. It is more appropriate for most “easier” problems where the convergence is relatively fast. User defined sampling functions must accept two arguments of n sampling points of dimension dim per call and output an array of s ampling points with shape n x dim. See SHGO.sampling_sobol for an example function.

Returns

res : OptimizeResult


The optimization result represented as a OptimizeResult object. Important attributes are: x the solution array corresponding to the global minimum, fun the function output at the global solution, xl an ordered list of local minima solutions, funl the function output at the corresponding local solutions, success a Boolean flag indicating if the optimizer exited successfully, message which describes the cause of the termination, nfev the total number of objective function evaluations including the sampling calls, nlfev the total number of objective function evaluations culminating from all local search optimisations, nit number of iterations performed by the global routine.