BITEOPT - Derivative-Free Optimization Method

Introduction

BITEOPT is a free open-source stochastic non-linear bound-constrained derivative-free optimization method (algorithm, heuristic, or strategy) for global optimization. The name "BiteOpt" is an acronym for "BITmask Evolution OPTimization".

The benefit of this method is a relatively high robustness: it can successfully optimize a wide range of multi-dimensional test functions. Another benefit is a low convergence time which depends on the complexity of the objective function. Hard (multi-modal) problems may require many optimization attempts to reach optimum.

Instead of iterating through different "starting guesses" to find optimum like in deterministic methods, this method requires optimization attempts with different random seeds. The stochastic nature of the method allows it to automatically "fall" into different competing minima on each attempt. If there are no competing minima in a function present (or the true/global minimum is rogue and cannot be detected), this method in absolute majority of attempts returns the same optimum.

BiteOpt uses self-optimization techniques making it objective function-agnostic. In its inner workings, BiteOpt uses objective function value's ranking, and not the actual value. BiteOpt is a multi-faceted example of a "probabilistic computing" system.

Python binding is available as a part of fcmaes library

Sponsoring

If you are regularly using BiteOpt in a commercial environment, you may consider donating/sponsoring the project. Please contact the author via aleksey.vaneev@gmail.com or info@voxengo.com. A solver for AMPL NL models is available commercially.

Comparison

This "black-box" optimization method was tested on 2000+ 1- to 60-dimensional optimization problems and performed well, and it successfully solves even 600-dimensional test problems found in some textbooks. But the main focus of the method is to provide fast solutions for computationally expensive "black-box" problems of medium dimensionality (up to 60).

This method was compared with the results of this paper (on 244 published C non-convex smooth problems, convex and non-convex non-smooth problems were not evaluated): Comparison of derivative-free optimization algorithms. This method was able to solve 76% of non-convex smooth problems in 10 attempts, 2500 iterations each. It comes 2nd in the comparison on non-convex smooth problems (see Fig.9 in the paper). With a huge iteration budget (up to 1 million) this method solves 97% of problems.

On a comparable test function suite and conditions outlined at this page: global_optimization (excluding several ill-defined and overly simple functions, and including several complex functions, use test2.cpp to run the test) this method's attempt success rate is >94% (with 100% of functions solved) while the average number of objective function evaluations is ~370.

At least in these comparisons, this method performs better than plain CMA-ES which is also a well-performing stochastic optimization method. As of version 2021.1, BiteOpt's "solvability" exceeds CMA-ES on synthetic function sets that involve random coordinate axis rotations and offsets (e.g., BBOB suite). BiteOptDeep (e.g., with M=8) considerably outperforms CMA-ES in "solvability".

As a matter of sport curiosity, BiteOpt is able to solve, in reasonable time, almost all functions proposed in classic academic literature on global optimization. This is quite a feat for a derivative-free method (not to be confused with large-scale analytic and gradient-based global optimization methods). Of course, BiteOpt is capable of more than that. If you have a reference to a function (with a known solution) published in literature that BiteOpt can't solve, let the author know.

BiteOpt (state at commit 124) took 2nd place (1st by sum of ranks) in BBComp2018-1OBJ-expensive competition track. Since the time of that commit the method improved in many aspects, especially in low-dimensional convergence performance. Commit 124 can be considered as "baseline effective" version of the method (it is also maximally simple), with further commits implementing gradual improvements, but also adding more complexity.

Also, BiteOpt (state at commit 256) took 2nd place (3rd by sum of ranks) in BBComp2019-1OBJ competition track.

CBiteOpt (biteopt.h)

BiteOpt optimization class. Implements a stochastic non-linear bound-constrained derivative-free optimization method. It maintains a cost-ordered population list of previously evaluated solutions that are evolved towards a lower cost (objective function value). On every iteration, the highest-cost solution in the list can be replaced with a new solution, and the list reordered. A population of solutions allows the method to space solution vectors apart from each other thus making them cover a larger parameter search space collectively. Beside that, a range of parameter randomization and the "step in the right direction" (Differential Evolution "mutation") operations are used that move the solutions into positions with a probabilistically lower objective function value.

Since version 2021.1 BiteOpt uses a companion optimizer - SpherOpt - which works independently and provides "reference points" to BiteOpt. Such companion improves BiteOpt's convergence properties considerably, especially when the parameter space is rotated. Since version 2021.15 BiteOpt uses an additional companion optimizer - NMSeqOpt - which increases diversity of generated solutions.

Since version 2021.3 BiteOpt became a self-optimizing method not requiring any fune-tuning from the user nor the author.

CBiteOptDeep (biteopt.h)

Deep optimization class. Based on an array of M CBiteOpt objects. This "deep" method pushes the newly-obtained solution to the random CBiteOpt object which is then optimized. This method, while increasing the convergence time, is able to solve complex multi-modal functions.

This method is most effective on complex functions, possibly with noisy fluctuations near the global solution, that are not very expensive to evaluate and that have a large iteration budget. Tests have shown that on smooth functions that have many strongly competing minima this "deep" method considerably increases the chance to find a global solution, relative to the CBiteOpt class, but still requires several attempts with different random seeds. When using this method, the required iteration budget usually increases by a factor of M^0.5, but the number of required optimization attempts usually decreases. In practice, it is not always possible to predict the convergence time increase of the CBiteOptDeep class, but increase does correlate to its M parameter. For some complex functions the use of CBiteOptDeep even decreases convergence time. For sure, the CBiteOptDeep class often produces better solutions than the CBiteOpt class.

Notes

BiteOpt is a completely self-optimizing method. It does not feature user-adjustable hyper-parameters. Even population size adjustments may not be effective.

It is usually necessary to run the optimization process several times, with different random seeds, since the process may get stuck at a local minimum. Running 10 times is a minimal general requirement. The required number of optimization attempts is usually proportional to the number of strongly competing minima in a function.

This method is hugely-probabilistic, and it depends on its initial state, which is selected randomly. In most cases it is more efficient to attempt to optimize with a new random seed than to wait for the optimization process to converge. Based on the results of optimization of the test-set, for 2-dimensional functions, it is reasonable to expect convergence in 800 iterations (in a successful attempt); for 10-dimensional functions, it is reasonable to expect convergence in 8000 iterations (harder functions may require more iterations to converge). Most classic 2-dimensional problems converge in 400 iterations or less, at 10^-6 precision. On average, every doubling of dimensions requires tripling of iteration budget. Complex multi-modal problems require a magnitudes larger iteration and attempt budget.

Each attempt may generate an equally-usable candidate solution (not necessarily having the least cost), permitting the researcher to select a solution from any attempt based on his/her own considerations. In this light, it may be incorrect to assume that least-performing attempts are "wasted". In practice, least-performing attempts may give more acceptable parameter values within the search space compared to the best-performing attempts.

Note that derivative-free optimization methods in general provide "asymptotic" solutions for complex functions. Thus it is reasonable to assume that BiteOpt gives an optimal solution with some implicit tolerance factor. Given a large enough function evaluation budget, BiteOpt usually does find an optimal solution which can be cross-checked with other solvers, but a solution of a new unexplored function must be treated as "asymptotically optimal".

Also note that in some problem areas like ESA GTOP problem suite the attempt budget should be as high as 1000 or more (beside using the BiteOptDeep depth of at least 6). At the same time, the iteration budget per attempt can be kept moderate (250000), compared to usual techniques used to solve these problems. Despite a large attempt budget, on a 8-core processor, this still allows one to get good (not necessarily best-known) solutions in a matter of minutes per problem.

Limitations

Rogue optimums may not be found by this method. A rogue optimum is an optimum that has a very small, almost undetectable area of descent, and is placed apart from other competing minima. The method favors minimum with a larger area of descent. The Damavandi test function is a perfect example of the limitation of this method (this test function is solved by this method, but requires a lot of iterations). In practice, however, rogue optimums can be considered as undesired outliers that rely on unstable parameter values: if such parameters are used in a real-world system that has a certain parameter value precision, a system may leave the "rogue" optimal regime easily. Another class of optimums the method cannot cope with well are "shadowed" optimums - the optimums that are located very close to opposite extremums.

To some degree, this method is immune to noise in the objective function. While this method was designed to be applied to continuous functions, it is immune to discontinuities, and it can solve problems that utilize parameter value rounding (integer parameters). This method usually can't acceptably solve high-dimensional continuous problems that are implicitly combinatorial (e.g., Perm, and Lennard-Jones atom clustering problems) as in such problems the global descent vanishes at some point and the method is left with an exponentially increasing number of local minima.

Also, problems with many competing minima without a pronounced global descent towards global minimum (e.g., Bukin N.6 problem) may not be solved acceptably as in most cases they require exhaustive search or a search involving knowledge of the structure of the problem. When the problem field requires one to locate such "rogue optimums", the best approach is to use a magnitudes larger attempt budget (preferrably using parallel computation like in the fcmaes library). With 1000 attempts and 250000 iterations per attempt budget, BiteOpt solves even the Bukin N.6 problem. It may seem excessive, but currently BiteOpt does not offer another way to solve such extremal multi-modal problems.

A difference between upper and lower parameter bound values should be specified in a way to cover a wider value range, in order to reduce boundary effects that may reduce convergence.

Tests have shown that in comparison to stochastic method like CMA-ES, BiteOpt's convergence time varies more from attempt to attempt. For example, on some problem CMA-ES's average convergence time may be 7000 iterations +/- 1000 while BiteOpt's may be 7000 +/- 3000. Such higher standard deviation is mostly a negative property if only a single optimization attempt is performed since it makes required iteration budget unpredictable. But if several attempts are performed, it is a positive property: it means that in some optimization attempts BiteOpt converges faster and may find a better optimum with the same iteration budget per attempt. Based on test2.cpp (2-dimensional) and test3.cpp (14-dimensional) test-sets, less than 0.9% of attempts require more than 3*sigma iterations, 54% of attempts require less than the mean. A typical probability distribution of percent of attempts/sigma is as follows (discretized, not centered around 0 because it deviates from the standard distribution, the mean corresponds to 0*sigma):

Traveling Salesman Problem (TSP)

BiteOptDeep is able to solve symmetric and asymmetric TSP problems even as large as 400-node ones, to within 3-8% of optimum. A comparison to a specialized TSP solver like Concorde is not reasonable to do (BiteOpt is much slower), but BiteOpt permits solving non-conventional or mixed-field (e.g., noisy, scheduled, clustered) discrete combinatorial problems.

To obtain node ordering when solving TSP (normal or clustered) problems, the parameters related to a corresponding cluster should be sorted in an ascending order. Each parameter corresponds to a node. As a result, BiteOpt "optimizes" the ranks of nodes in a cluster. Minimal parameter bounds can be all set to 0, while maximal bounds can be all set to 1.

Note that iteration and attempt budgets are needed to be quite large, to reach good results or an optimum.

Constraint Programming

Mixed integer programming can be achieved by using rounded parameter values in the objective function. Note that using categorical variables may not be effective, because they require combinatorial search. Binary variables may be used, in small quantities (otherwise the problem usually transforms into a combinatorial problem as well).

Equality and non-equality constraints - both linear and non-linear - can be implemented as penalties. The author has found a general effective approach to apply constraints via penalties. While penalties are not well-regarded in research community, BiteOpt handles constraint penalties extremely well, but usually requiring a large iteration budget (suitable for inexpensive objective functions).

In the code below, n_con is the number of constraints, con_notmet is the number of constraints not meeting tolerances, and the pn[] is the array of positive penalty values for each constraint; a penalty value should be set to 0, if it meets the tolerance (a penalty value should be offseted by tolerance factor to make smooth approach towards 0). For derivative-free methods, a suggested constraint tolerance is 10^-4, but a more common 10^-6 can be also used; lower values are not advised for use. Models with up to 200 constraints, both equalities and non-equalities, were tested with this method. In practice, on a large set of problems, this method finds a feasible solution in up to 97% of cases (with 20-30 attempts per problem).

real_value = cost;

if( con_notmet > 0 )
{
    const double ps = 1.0 + 1.0 / n_con;
    double pns = 0.0;

    for( int i = 0; i < n_con; i++ )
    {
        const double v = pn[ i ];
        pns = ( 1.0 + pns ) * ps + ( v + v * v + v * v * v ) * 0.33333;
    }

    cost += 1e10 * pns;
}

In essence, this approach transforms each penalty value into a cubic penalty value, places each penalty value into its own "stratum" (via "1.0" offset and "ps" multiplier), and also applies a "barrier value". The barrier value (1e10) is suitably large for most practical constraint programming problems.

See constr.cpp for an example of constraint programming. constr2.cpp is an example of non-linear constraint programming with both equalities and non-equalities. To effectively solve constraint programming problems, the CBiteOptDeep class should be used, with M=6 or higher.

It is not advisable to use constraints like (x1-round(x1)=0) for non-binary integer variables, commonly used in model libraries to force integer values, as such constraint formulation does not provide a useful global gradient. Instead, direct rounding should be used on integer variables.

BiteOpt is able to solve binary combinatorial problems, if the cost function is formulated as a sum of differences between bit values and continuous variables in the range [0; 1] - these differences can be used as usual constraints while binary value equality tolerance can be set to as low as 10^-12.

Multi-Objective Optimization

BiteOpt does not offer MOO "out of the box". However, BiteOpt can successfully solve MOO problems via direct optimization of hypervolume of a set of points. This approach requires a hypervolume tracker which keeps track of a certain number of improving solutions, and updates its state (and hypervolume estimate) on each objective function evaluation (optcost). The approach is demonstrated in fcmaes tutorial - quantumcomm.py.

Convergence Proof

Considering the structure of the method and the fact that on every iteration only improving solutions are accepted into the population, with ever decreasing upper bound on the objective function value, it is logically impossible for the method to be divergent. While it is strictly non-divergent, the formal proof of ability of the method to converge is complicated, and should be at least as good as partly random search and partly Differential Evolution.

Tested Uses

This optimization method was tested for the following applications beside synthetic benchmarking:

• Hyper-parameter optimization of complex non-linear black-box systems. Namely, AVIR image resizing algorithm's hyper-parameters, digital audio limiter algorithm's parameters.

• Non-linear least-squares problems, see the calcHougen and calcOsborne functions in the testfn.h file for example problems, and also the suite of NIST/ITL StRD problems in test_nist.cpp.

• BiteOptDeep was successfuly used for direct search of optimal short symmetric FIR filters. Namely, in r8brain-free-src sample rate converter.

• BiteOpt is featured as an optimizer in M-Star CFD physical modeling system

• A variety of tutorial problems in fcmaes library: 5G network planning, Discrete clustering, Cryptocurrency trading, Employee scheduling.

BiteOpt is also referenced in these research papers:

• Password Strength Signaling: A Counter-Intuitive Defense Against Password Cracking, Springer

• The CIP2A-TOPBP1 axis safeguards chromosome stability and is a synthetic lethal target for BRCA-mutated cancer, Nature Cancer

• Automating Building Damage Reconnaissance to Optimize Drone Mission Planning for Disaster Response, ASCE Library

• Cost-Asymmetric Memory Hard Password Hashing, ACM DL

• Quantum Simulation of Spin-Boson Models with Structured Bath, arXiv

Examples

Use the example.cpp program to see the basic usage example of C++ interface.

The example2.cpp program is a usage example of a simple C-like function biteopt_minimize(). This is a minimization test for Hougen-Watson model for reaction kinetics (non-linear least squares problem).

int biteopt_minimize( const int N, biteopt_func f, void* data,
    const double* lb, const double* ub, double* x, double* minf,
    const int iter, const int M = 1, const int attc = 10,
    const int stopc = 0, biteopt_rng rf = 0, void* rdata = 0,
    double* f_minp = 0 )

N     The number of parameters in an objective function.
f     Objective function.
data  Objective function's data.
lb    Lower bounds of obj function parameters, should not be infinite.
ub    Upper bounds of obj function parameters, should not be infinite.
x     Minimizer.
minf  Minimizer's value.
iter  The number of iterations to perform in a single attempt.
      Corresponds to the number of obj function evaluations that are performed.
M     Depth to use, 1 for plain CBiteOpt algorithm, >1 for CBiteOptDeep
      algorithm. Expected range is [1; 36]. Internally multiplies "iter"
      by sqrt(M).
attc  The number of optimization attempts to perform.
stopc Stopping criteria (convergence check). 0: off, 1: 128*N, 2: 256*N.
rf    Random number generator function; 0: use the default BiteOpt PRNG.
      Note that the external RNG should be seeded externally.
rdata Data pointer to pass to the "rf" function.
f_minp If non-zero, a pointer to the stopping value: optimization will
      stop when this objective value is reached.

This function returns the total number of function evaluations performed;
useful if the "stopc>0" and/or "f_minp" were used.

test2.cpp is a convergence test for all available functions. Performs many optimization attempts on all functions. Prints various performance information, including per-function and per-attempt success rates.

test3.cpp is a convergence test for multi-dimensional functions with random axis rotations and offsets.

test4.cpp is a convergence test for multi-dimensional functions without randomization.

constr.cpp, constr2.cpp, and constr3.cpp programs demonstrate the use of constraint penalties.

Development

There are several things that were discovered that may need to be addressed in the future:

1. Parallelization of BiteOpt algorithm is technically possible, but may be counter-productive (increases convergence time considerably). It is more efficient to run several optimizers in parallel with different random seeds. Specifically saying, it is possible (tested to be working on some code commits before May 15, 2018) to generate series of candidate solutions, evaluate them in parallel, and then update optimizer's state before generating a new batch of candidate solutions. Later commits have changed the algorithm to a form less suitable for such parallelization.

2. The method uses "short-cuts" which can be considered as "tricks" (criticized in literature) which are non-universal, and reduce convergence time out of proportion for many known test functions. These "short-cuts" are not critically important to method's convergence properties, but they reduce convergence time even for functions that do not have minimum at a point where all arguments are equal. It just often happens that such "short-cuts" provide useful "reference points" to the method. Removing these "short-cuts" will increase average convergence time of the method, but in most cases won't impact method's ability to find a global solution. "Short-cuts" are used only in 4% of objective function evaluations, on average.

Method's Philosophy

BiteOpt is an evolutionary optimization method. Unlike many established optimization methods like CMA-ES where new populations are generated on each iteration, with or without combining with the previous generation, BiteOpt keeps and updates a single main population of solutions, at any given time. A new solution either replaces a worst solution or is discarded. In common terms, it means that population has some fixed "living space" which is only available to the best fit (least cost) solutions. Structurally, this is similar to a natural evolutionary environment which usually offers only a limited "living space" to its members. Least fit members have little chance to stay in this "living space".

BiteOpt uses several functions to generate new solutions, each function taking various information from various internal populations. These functions are used in a probabilistic manner without any predefined preference.

BiteOptDeep implements evolutionary method which can be seen in society and nature: exchange of solutions between independent populations. Such exchange permits search of better solutions in a teamwork of sufficiently diverse members; it also reduces time (but not human-hours) to find a better solution. This method is a model of Swiss presidency rotation (each independent population represents an independent human). Note that the very best solution found by a member is not shared with other members as to not speed-up the convergence unnecessarily.

The author did not originally employ results and reasoning available in papers on Differential Evolution (DE). Author's use of DE operation is based on understanding that it provides an implicit gradient information. A candidate solution is generated as a sum of best solution and a difference between a random and the worst solution. Such difference (between a random and the worst solution) generates a probabilistically correct step towards the minimum of a function, relative to a better solution. Due to this understanding, it is impossible to employ various DE variants in BiteOpt, only the difference between high rank and low rank solutions generates a valuable information; moreover, only a difference multiplied by a factor of 0.5 works in practice. Since BiteOpt does not use a classic random crossover in its DE-alike operations, the used approach is closer to an intermix of Nelder-Mead ("reduction") and DE (multi-vector "mutation").

BiteOpt is more like a stochastic meta-method, and it is incorrect to assume it leans towards some specific optimizer class: for example, it won't work acceptably if only DE-alike solution generators are used by it. BiteOpt encompasses Differential Evolution, Nelder-Mead, author's original SpherOpt, "bitmask inversion", and "bit mixing" (genetic) solution generators. An initial success with the "bitmask inversion" operation (coupled with a stochastic "move" operation it looks quite a lot like a random search) was the main driver for BiteOpt's further development.

Method's Description

Overview

A cost-ordered population of previous solutions is maintained. A solution is an independent parameter vector which can be used to generate/compose a new candidate solution by a selected solution generator. On every iteration, the method utilizes a probabilistically-chosen candidate solution generator. At the start, the solution vectors are initialized at the center of the search space, using Gaussian sampling.

Beside the main population, the method keeps several "parallel populations" that are updated on the basis of proximity of a candidate solution to a given population's centroid. As a result, these populations tend to slightly diverge from both each other and the main population.

Parameter values are internally normalized to [0; 1] range and, to stay in this range, are wrapped in a special manner before each function evaluation. The method uses an alike of a probabilistic state-automata (by means of "selectors") to switch between algorithm flow-paths, depending on the candidate solutions' acceptance on previous iterations. Each selector represents a superposition of flow-paths, with each flow-path initially being equally-probable. Depending on the acceptance or rejection of a newly-generated candidate solution, the selector is updated accordingly, and the "probabilistic weight" of a recently used flow-path is adjusted. This approach increases the number of acceptable solutions, and produces a smoother descent.

In many instances candidate solution generators use the squared, power'ed or logarithmic random variable to obtain solution's index: this has an effect of giving more weight to better solutions.

With some probability, an independent, algorithmically different, parallel optimizer is engaged whose solution is evaluated for inclusion into the population. The solutions of parallel optimizers are kept in additional independent populations, and they can be used by the solution generators.

After each objective function evaluation, the highest-cost previous solution is replaced, if new solution's cost meets the upper bound cost constraint.

Solution Generators

Note that most solution generators can be used on their own (with some minor tweaks, especially population size) as lower-quality solvers, but some generators can't work well on their own (they work due to synergistic effects), and are used to increase the diversity of solution approaches. In some instances a generator may produce an acceptable solution only once per 50 calls, but this solution may make a big difference in a long run.

Some solution generators work well for discrete problems, but may perform poorly for continuous optimization; some are good for both. In the main 4-method flow-path, paths 0, 2, and 3 feature solution generators that solve discrete problems well - they are staple solution generators, while path 1 includes additional, less important, generators. An ability to solve discrete problems improves method's tolerance to noise.

The availability of many solution generators seems to be essential for solving discrete and noisy problems. While generators 1, 2, 3 can be considered "exploitation" generators as they provide faster convergence and better solution values, all other generators are "exploration" generators.

1. A single (or all) parameter value randomization is performed using the "bitmask inversion" operation (which is approximately equivalent to v=1-v operation in normalized parameter space). Below, i is either equal to rand(1, N) or in the range [1; N], depending on the Allp probability. >> is a bit shift-right operation, IntMantBits is a constant equal to 58, MantSizeSh is a fixed parameter that specifies bit shift operation's range. Actual implementation is more complex as it uses the average of two such operations.

$$ mask=(2^{IntMantBits}-1)\gg \lfloor rand(0\ldots1)^4\cdot MantSizeSh\rfloor $$

$$ x_\text{new}[i] = \frac{\lfloor x_\text{best}[i]\cdot 2^{IntMantBits} \rfloor \bigotimes mask }{2^{IntMantBits}} $$

Plus, with 1-1/Dims probability the move around a random previous solution is performed, utilizing a TPDF random value (the difference between two solutions represents estimation of standard deviation). This operation is performed twice.

$$ x_\text{new}[i]=x_\text{new}[i]-rand_{TPDF}(-1\ldots1)\cdot CentSpan\cdot (x_\text{new}[i]-x_\text{rand}[i]) $$

2. The "step in the right direction" operation. Uses the random previous solution, chosen best and worst solutions, plus a difference of two other random solutions. This is conceptually similar to Differential Evolution's "mutation" operation. The worst solution is selected symmetrically relative to the chosen best solution.

$$ x_\text{new}=x_\text{best}+\frac{(x_\text{rand}-x_\text{worst}+ (x_\text{rand2}-x_\text{rand3}))}{2} $$

3. Involves a better solution, centroid vector, and a random worse solution.

$$ x_\text{new}[i]=x_\text{best}[i]+x_\text{rand}[i]+(-1)^{s}(x_\text{cent}[i]- x_\text{rand}[i]), \quad i=1,\ldots,N,\ \quad s\in{1,2}= (\text{rand}(0\ldots1)<0.5 ? 1:2) $$

4. The "entropy bit mixing" method. This method mixes (XORs) parameter values represented as raw bit strings drawn from an odd number of parameter vectors. Probabilistically, such composition creates a new random parameter vector, with an overwhelming number of bits being common to the better-performing solutions, and a fewer number of bits without fitness certainty.

5. A novel "Randomized bit crossing-over" candidate solution generation method. Effective, but on its own cannot stand coordinate system offsets, converges slowly. Completely mixes bits of two randomly-selected solutions, plus changes 1 random bit. Uses a random mix-mask. This method is similar to a biological DNA crossing-over, but on a single-bit scale.

6. The "short-cut" parameter vector generation.

$$ z=x_\text{best}[\text{rand}(1\ldots N)] $$

$$ x_\text{new}[i]=z, \quad i=1,\ldots,N $$

7. A solution generator that randomly combines solutions from the main and "old" populations. Conceptually, it can be called a weighted-random crossover that combines solutions from diverse sources.

8. Solution generator that is DE-alike at its base. It calculates a centroid of a number of best solutions, and then applies "mutation" operation between the centroid and the solutions, using a random multiplier. This approach is similar to the "move" operation of generator 1.

9. The "water drain" solution generator. It moves a random (better) solution away or towards a worse solution, using a fixed step multiplier. This is reminiscent of a process of water drainage when a higher-elevation molecule excerts a gravity-induced pressure on a lower-elevation molecule, with two possible outcomes per parameter: either the lower-elevation molecule moves further down or bounces back upper.

10. Solution generator derived from SpherOpt's converging hyper-spheroid method.

11. Stochastic PSO-alike solution generator. Moves a randomly-selected existing solution towards a better solution, and at the same times makes a move in a random direction having a magnitude derived as a distance between a better and worse solutions (limits the magnitude to the current basin of solutions).

12. Solution generator that estimates population's standard deviation using better and worse solutions, and then generates a solution using Gaussian sampling around centroid.

13. Solution generator that applies Differential Evolution in real parameter value space, in a randomized fashion: each parameter value receives a DE operation value of a randomly-chosen parameter.

SpherOpt

This is a "converging hyper-spheroid" optimization method (or hyper-sphere, depending on optimization space's bounds). While it is not as effective as, for example, CMA-ES, it also stands parameter space scaling, offsetting, and rotation well. Since version 2021.1 it is used as a companion (parallel optimizer) to BiteOpt, with excellent results.

This method is in parts similar to SMA-ES, but instead of keeping track of per-parameter sigmas, covariance matrix, and using Gaussian sampling, SpherOpt simply selects random points on a hyper-spheroid (with a bit of added jitter at lower dimensions), which eventually converges to a point. This makes the method very computationally-efficient, but at the same time provides immunity to coordinate axis rotations.

This method uses the same self-optimization technique as BiteOpt which is, however, not a vital element of the method.

MiniBiteOpt

This solver is a minimized version of BiteOpt. This version incorporates the most effective solution generators reminiscent of early BiteOpt versions. This solver is used as an additional parallel optimizer in BiteOpt.

NMSeqOpt

The CNMSeqOpt class implements sequential Nelder-Mead simplex method with the "stall count" tracking. This optimizer is used as an additional parallel optimizer in BiteOpt.

SMA-ES

This is a working optimization method called "SigMa Adaptation Evolution Strategy". It has the same programmatic interface as the CBiteOpt class, so it can be easily used in place of CBiteOpt.

SMA-ES is based on the same concept as CMA-ES, but performs vector sigma adaptation. SMA-ES performs covariance matrix update like CMA-ES, but it is a simple linear update using "leaky integrator" averaging filtering, not adaptation. SMA-ES algorithm's operation is based on principles of control signals.

The main difference to CMA-ES is that per-parameter sigmas are updated using these elements:

1. Sigma auto-adapts due to weighted parameter covariance calculation. Better fit solutions have more influence over expansion or contraction of the sigma.

2. SMA-ES approximates the "geometry" of the sample distribution. It ranges from "spherical" to "needle" geometry (represented by a continuous spc variable). When geometry is spherical, covariance matrix update filter is tuned to an increased frequency (CovUpdFast instead of CovUpdSlow).

3. An asymmetry is introduced to the Gaussian sampling function, depending on the centroid step size. Distribution is expanded in the direction of the step and contracted in the opposite direction.

4. On every update, all per-parameter sigmas are contracted (multiplied) by the SigmaMulBase coefficients, depending on sphericity. Additionally, overly-contracted sigmas are expanded by the SigmaMulExp coefficient.

In overall, SMA-ES is a completely self-adaptive method. It has several hyper-parameters that do not depend on problem's dimensionality.

Population size formula in SMA-ES is fixed to 13+Dims: according to tests, in average it suits all dimensionalities. Of course, particular problems may converge better/faster with a lower or higher population size. The number of objective function evaluations is twice the population size per sample distribution update (best fit solutions enter the population): this aspect is controlled via the EvalFac parameter, which adjusts method's overhead with only a minor effect on convergence property. Method's typical observed complexity is O(N^1.6).

DEOpt

The CDEOpt class implements a Differential Evolution-alike DFO solver, but in the population-handling framework of BiteOpt. "DE/best-2/3/bit". Mutation parameter is fixed, equals to 0.25. Instead of a crossover, the method uses randomization. Population size is equal to 30*Dims, by default. Population is initialized with Gaussian sampling.

Citing

@misc{biteopt2024,
    author = {Aleksey Vaneev},
    title = {{BITEOPT - Derivative-free global optimization method}},
    note = {C++ source code, with description and examples},
    year = {2024},
    publisher = {GitHub},
    journal = {GitHub repository},
    howpublished = {Available at \url{https://github.com/avaneev/biteopt}},
}