Optimization algorithms for scalar and vector functions. More...

Data Structures
struct	FRPRConfig
	Configuration for FRPR conjugate gradient method. More...

struct	GlobalConfig
	Configuration for global optimization. More...

struct	NewtonConfig
	Configuration for Newton minimization with Hessian. More...

struct	PSOConfig
	Configuration for Particle Swarm Optimization. More...

struct	PSOSeedParticle
	Seed particle for PSO initialization. More...

struct	PSOStats
	Statistics collected during PSO execution. More...

struct	UzawaConfig
	Configuration for Uzawa algorithm. More...

struct	UzawaResult
	Result for Uzawa constrained optimization. More...

Enumerations
enum class	ConjugateGradientFormula { FletcherReeves , PolakRibiere , HestenesStiefel , DaiYuan }
	Conjugate gradient formula selection. More...

enum class	PSOInitMode { RandomOnly , SeededOnly , SeededPlusRandom }
	Initialization mode for PSO particles. More...

enum class	PSOBoundaryMode { Clamp , Reflect , Wrap }
	Boundary handling mode for particles leaving the search space. More...

enum class	PSOInertiaSchedule { Constant , LinearDecay }
	Inertia weight schedule. More...

enum class	GlobalStrategy { PSO , MultiStart , PSOHybrid , DifferentialEvolution }
	Global optimization strategy selection. More...

Functions
template<typename Function >
ScalarResult	Brent (Function &theFunc, double theLower, double theUpper, const Config &theConfig=Config())
	Brent's method for 1D minimization. Combines golden section search with parabolic interpolation. Guaranteed to converge for unimodal functions within the given interval.

template<typename Function >
ScalarResult	Golden (Function &theFunc, double theLower, double theUpper, const Config &theConfig=Config())
	Golden section search for 1D minimization. Simpler than Brent but with guaranteed linear convergence. Does not attempt parabolic interpolation.

template<typename Function >
ScalarResult	BrentWithBracket (Function &theFunc, double theGuess, double theStep=1.0, const Config &theConfig=Config())
	Brent's method with automatic bracket search. First attempts to bracket a minimum, then applies Brent's method.

template<typename Function >
VectorResult	Powell (Function &theFunc, const math_Vector &theStartingPoint, const Config &theConfig=Config())
	Powell's conjugate direction method for N-dimensional minimization. A gradient-free optimization algorithm that uses conjugate directions.

template<typename Function >
VectorResult	PowellWithDirections (Function &theFunc, const math_Vector &theStartingPoint, const math_Matrix &theInitialDirections, const Config &theConfig=Config())
	Powell's method with custom initial directions. Allows specifying the initial direction set instead of coordinate axes.

template<typename Function >
VectorResult	BFGS (Function &theFunc, const math_Vector &theStartingPoint, const Config &theConfig=Config())
	BFGS (Broyden-Fletcher-Goldfarb-Shanno) quasi-Newton method. One of the most effective algorithms for smooth unconstrained optimization.

template<typename Function >
VectorResult	BFGSNumerical (Function &theFunc, const math_Vector &theStartingPoint, double theGradStep=1.0e-8, const Config &theConfig=Config())
	BFGS with numerical gradient. Uses central differences to approximate gradient when analytical gradient is not available.

template<typename Function >
VectorResult	LBFGS (Function &theFunc, const math_Vector &theStartingPoint, int theMemorySize=10, const Config &theConfig=Config())
	L-BFGS (Limited-memory BFGS) for large-scale optimization. Uses only the m most recent {s, y} pairs instead of full Hessian. Memory: O(m*n) instead of O(n^2).

template<typename Function >
VectorResult	BFGSBounded (Function &theFunc, const math_Vector &theStartingPoint, const math_Vector &theLowerBounds, const math_Vector &theUpperBounds, const Config &theConfig=Config())
	BFGS with bound constraints (box constraints).

template<typename Function >
VectorResult	FRPR (Function &theFunc, const math_Vector &theStartingPoint, const FRPRConfig &theConfig=FRPRConfig())
	Fletcher-Reeves-Polak-Ribiere conjugate gradient method.

template<typename Function >
VectorResult	FRPRNumerical (Function &theFunc, const math_Vector &theStartingPoint, double theGradStep=1.0e-8, const FRPRConfig &theConfig=FRPRConfig())
	FRPR with numerical gradient. Uses central differences when analytical gradient is not available.

template<typename Function >
VectorResult	Newton (Function &theFunc, const math_Vector &theStartingPoint, const NewtonConfig &theConfig=NewtonConfig())
	Newton's method for N-dimensional minimization using Hessian.

template<typename Function >
VectorResult	NewtonModified (Function &theFunc, const math_Vector &theStartingPoint, const NewtonConfig &theConfig=NewtonConfig())
	Modified Newton's method with automatic Hessian regularization. Adds diagonal elements to ensure positive definiteness using an adaptive regularization strategy.

template<typename Function >
VectorResult	NewtonNumericalHessian (Function &theFunc, const math_Vector &theStartingPoint, double theHessStep=1.0e-6, const NewtonConfig &theConfig=NewtonConfig())
	Newton's method with numerical Hessian. Computes Hessian using finite differences when analytical Hessian is not available.

template<typename Function >
VectorResult	NewtonNumerical (Function &theFunc, const math_Vector &theStartingPoint, double theGradStep=1.0e-8, double theHessStep=1.0e-6, const NewtonConfig &theConfig=NewtonConfig())
	Newton's method with fully numerical derivatives. Computes both gradient and Hessian using finite differences.

template<typename Function >
VectorResult	NewtonBounded (Function &theFunc, const math_Vector &theStartingPoint, const math_Vector &theLowerBounds, const math_Vector &theUpperBounds, const NewtonConfig &theConfig=NewtonConfig())
	Newton's method with bound constraints.

template<typename Function >
void	PolishCoordinateWise (Function &theFunc, math_Vector &thePosition, double &theValue, const math_Vector &theLowerBounds, const math_Vector &theUpperBounds, double theTolerance, int theMaxPolishEvals, int &theEvalCount)
	Coordinate-wise polishing using Brent's 1D minimization. For each dimension, performs Brent's method directly on the single coordinate, modifying only one element of the position vector per evaluation (zero-allocation).

template<typename Function >
VectorResult	PSO (Function &theFunc, const math_Vector &theLowerBounds, const math_Vector &theUpperBounds, const PSOConfig &theConfig, const NCollection_DynamicArray< PSOSeedParticle > theSeeds, PSOStats theStats=nullptr)
	Particle Swarm Optimization for global minimization.

template<typename Function >
VectorResult	PSO (Function &theFunc, const math_Vector &theLowerBounds, const math_Vector &theUpperBounds, const PSOConfig &theConfig=PSOConfig())
	Particle Swarm Optimization for global minimization (basic overload).

template<typename Function >
VectorResult	DifferentialEvolution (Function &theFunc, const math_Vector &theLowerBounds, const math_Vector &theUpperBounds, const GlobalConfig &theConfig=GlobalConfig())
	Differential Evolution algorithm for global optimization.

template<typename Function >
VectorResult	MultiStart (Function &theFunc, const math_Vector &theLowerBounds, const math_Vector &theUpperBounds, const GlobalConfig &theConfig=GlobalConfig())
	Multi-start local optimization.

template<typename Function >
VectorResult	GlobalMinimum (Function &theFunc, const math_Vector &theLowerBounds, const math_Vector &theUpperBounds, const GlobalConfig &theConfig=GlobalConfig())
	Unified global optimization interface.

template<typename Function >
VectorResult	GlobalMinimum (Function &theFunc, const math_Vector &theLowerBounds, const math_Vector &theUpperBounds, const GlobalConfig &theConfig, const PSOConfig thePSOConfig, const NCollection_DynamicArray< PSOSeedParticle > theSeeds=nullptr, PSOStats *theStats=nullptr)
	Unified global optimization interface with PSO-specific options.

UzawaResult	Uzawa (const math_Matrix &theCont, const math_Vector &theSecont, const math_Vector &theStartingPoint, int theNce, int theNci, const UzawaConfig &theConfig=UzawaConfig())
	Solve constrained least squares using Uzawa algorithm.

UzawaResult	UzawaEquality (const math_Matrix &theCont, const math_Vector &theSecont, const math_Vector &theStartingPoint, const UzawaConfig &theConfig=UzawaConfig())
	Solve constrained least squares with equality constraints only.

Detailed Description

Optimization algorithms for scalar and vector functions.

Enumeration Type Documentation

◆ ConjugateGradientFormula

enum class MathOpt::ConjugateGradientFormula

strong

Conjugate gradient formula selection.

Enumerator
FletcherReeves	beta = g_new^T g_new / g^T g (original, guaranteed descent)
PolakRibiere	beta = g_new^T (g_new - g) / g^T g (often faster, may need restarts)
HestenesStiefel	beta = g_new^T (g_new - g) / d^T (g_new - g)
DaiYuan	beta = g_new^T g_new / d^T (g_new - g)

◆ GlobalStrategy

enum class MathOpt::GlobalStrategy

strong

Global optimization strategy selection.

Enumerator
PSO	Particle Swarm Optimization only.
MultiStart	Multiple local optimizations from random starts.
PSOHybrid	PSO followed by local refinement.
DifferentialEvolution	Differential Evolution algorithm.

◆ PSOBoundaryMode

enum class MathOpt::PSOBoundaryMode

strong

Boundary handling mode for particles leaving the search space.

Enumerator
Clamp	Clamp to bound, reverse velocity with -0.5 damping (default)
Reflect	Reflect coordinate, damp velocity by -0.5.
Wrap	Periodic wrap into bounds.

◆ PSOInertiaSchedule

enum class MathOpt::PSOInertiaSchedule

strong

Inertia weight schedule.

Enumerator
Constant	Fixed omega (default)
LinearDecay	Linear decay from Omega to OmegaMin over iterations.

◆ PSOInitMode

enum class MathOpt::PSOInitMode

strong

Initialization mode for PSO particles.

Enumerator
RandomOnly	All particles randomly initialized (default)
SeededOnly	Initialize from seed particles only.
SeededPlusRandom	Seed particles + remaining random.

Function Documentation

◆ BFGS()

template<typename Function >

VectorResult MathOpt::BFGS	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		const Config &	theConfig = Config() )

BFGS (Broyden-Fletcher-Goldfarb-Shanno) quasi-Newton method. One of the most effective algorithms for smooth unconstrained optimization.

Algorithm:

Start with initial Hessian approximation H (usually identity)
Compute search direction p = -H * gradient
Perform line search to find step size alpha satisfying Wolfe conditions
Update x = x + alpha * p
Update Hessian approximation using BFGS formula
Repeat until convergence

The BFGS update maintains positive definiteness of H if started with a positive definite matrix and using proper line search.

Advantages:

Superlinear convergence near minimum
Self-correcting Hessian approximation
No need to compute actual Hessian

Template Parameters

Function

type with:

Value(const math_Vector&, double&) for function value
Gradient(const math_Vector&, math_Vector&) for gradient

Parameters

theFunc	function object with value and gradient
theStartingPoint	initial guess
theConfig	solver configuration

Returns: result containing minimum location and value

◆ BFGSBounded()

template<typename Function >

VectorResult MathOpt::BFGSBounded	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		const math_Vector &	theLowerBounds,
		const math_Vector &	theUpperBounds,
		const Config &	theConfig = Config() )

BFGS with bound constraints (box constraints).

Minimizes f(x) subject to theLowerBounds <= x <= theUpperBounds. Uses projected gradient approach:

After each step, clamp x to bounds
Zero gradient components at bounds where they point outward

Template Parameters

Function type with Value and Gradient methods

Parameters

theFunc	function object
theStartingPoint	initial guess
theLowerBounds	lower bounds for each variable
theUpperBounds	upper bounds for each variable
theConfig	solver configuration

Returns: result containing minimum location and value

◆ BFGSNumerical()

template<typename Function >

VectorResult MathOpt::BFGSNumerical	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		double	theGradStep = 1.0e-8,
		const Config &	theConfig = Config() )

BFGS with numerical gradient. Uses central differences to approximate gradient when analytical gradient is not available.

Template Parameters

Function type with Value(const math_Vector&, double&) method only

Parameters

theFunc	function object
theStartingPoint	initial guess
theGradStep	step size for numerical gradient (default 1e-8)
theConfig	solver configuration

Returns: result containing minimum location and value

◆ Brent()

template<typename Function >

ScalarResult MathOpt::Brent	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		const Config &	theConfig = Config() )

Brent's method for 1D minimization. Combines golden section search with parabolic interpolation. Guaranteed to converge for unimodal functions within the given interval.

Algorithm:

Maintain bracket [a, b] with interior point x where f(x) < f(a), f(x) < f(b)
Try parabolic interpolation using three points
If parabolic step is rejected, use golden section step
Update bracket and repeat until convergence

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to minimize
theLower	lower bound of search interval
theUpper	upper bound of search interval
theConfig	solver configuration

Returns: result containing minimum location and value

◆ BrentWithBracket()

template<typename Function >

ScalarResult MathOpt::BrentWithBracket	(	Function &	theFunc,
		double	theGuess,
		double	theStep = 1.0,
		const Config &	theConfig = Config() )

Brent's method with automatic bracket search. First attempts to bracket a minimum, then applies Brent's method.

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to minimize
theGuess	initial guess
theStep	initial step size for bracket search
theConfig	solver configuration

Returns: result containing minimum location and value

◆ DifferentialEvolution()

template<typename Function >

VectorResult MathOpt::DifferentialEvolution	(	Function &	theFunc,
		const math_Vector &	theLowerBounds,
		const math_Vector &	theUpperBounds,
		const GlobalConfig &	theConfig = GlobalConfig() )

Differential Evolution algorithm for global optimization.

DE is a stochastic, population-based optimization algorithm. It uses mutation, crossover, and selection operations to evolve a population of candidate solutions.

Template Parameters

Function type with Value(const math_Vector&, double&) method

Parameters

theFunc	function to minimize
theLowerBounds	lower bounds
theUpperBounds	upper bounds
theConfig	optimization configuration

Returns: result containing best solution found

◆ FRPR()

template<typename Function >

VectorResult MathOpt::FRPR	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		const FRPRConfig &	theConfig = FRPRConfig() )

Fletcher-Reeves-Polak-Ribiere conjugate gradient method.

Memory-efficient alternative to BFGS for large-scale optimization. Uses only O(n) storage compared to O(n^2) for BFGS.

Algorithm:

Compute gradient g at current point
First iteration: search direction p = -g
Perform line search along p
Compute new gradient g_new
Update: beta = (g_new . g_new) / (g . g) [Fletcher-Reeves] or beta = (g_new . (g_new - g)) / (g . g) [Polak-Ribiere]
New direction: p = -g_new + beta * p
Restart with steepest descent if beta < 0 or periodically
Repeat until convergence

Template Parameters

Function

type with:

Value(const math_Vector&, double&) for function value
Gradient(const math_Vector&, math_Vector&) for gradient

Parameters

theFunc	function object with value and gradient
theStartingPoint	initial guess
theConfig	solver configuration

Returns: result containing minimum location and value

◆ FRPRNumerical()

template<typename Function >

VectorResult MathOpt::FRPRNumerical	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		double	theGradStep = 1.0e-8,
		const FRPRConfig &	theConfig = FRPRConfig() )

FRPR with numerical gradient. Uses central differences when analytical gradient is not available.

Template Parameters

Function type with Value(const math_Vector&, double&) method only

Parameters

theFunc	function object
theStartingPoint	initial guess
theGradStep	step size for numerical gradient
theConfig	solver configuration

Returns: result containing minimum location and value

◆ GlobalMinimum() [1/2]

template<typename Function >

VectorResult MathOpt::GlobalMinimum	(	Function &	theFunc,
		const math_Vector &	theLowerBounds,
		const math_Vector &	theUpperBounds,
		const GlobalConfig &	theConfig,
		const PSOConfig *	thePSOConfig,
		const NCollection_DynamicArray< PSOSeedParticle > *	theSeeds = nullptr,
		PSOStats *	theStats = nullptr )

Unified global optimization interface with PSO-specific options.

For PSO and PSOHybrid strategies, uses the provided PSO configuration, seed particles, and stats output. For other strategies, these are ignored.

Template Parameters

Function type with Value(const math_Vector&, double&) method

Parameters

theFunc	function to minimize
theLowerBounds	lower bounds for each variable
theUpperBounds	upper bounds for each variable
theConfig	solver configuration
thePSOConfig	optional PSO-specific configuration (overrides auto-generated)
theSeeds	optional seed particles for PSO initialization
theStats	optional PSO statistics output

Returns: result containing best solution found

◆ GlobalMinimum() [2/2]

template<typename Function >

VectorResult MathOpt::GlobalMinimum	(	Function &	theFunc,
		const math_Vector &	theLowerBounds,
		const math_Vector &	theUpperBounds,
		const GlobalConfig &	theConfig = GlobalConfig() )

Unified global optimization interface.

Selects appropriate algorithm based on configuration and provides a consistent interface for all global optimization methods.

Template Parameters

Function type with Value(const math_Vector&, double&) method

Parameters

theFunc	function to minimize
theLowerBounds	lower bounds for each variable
theUpperBounds	upper bounds for each variable
theConfig	solver configuration

Returns: result containing best solution found

◆ Golden()

template<typename Function >

ScalarResult MathOpt::Golden	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		const Config &	theConfig = Config() )

Golden section search for 1D minimization. Simpler than Brent but with guaranteed linear convergence. Does not attempt parabolic interpolation.

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to minimize
theLower	lower bound of search interval
theUpper	upper bound of search interval
theConfig	solver configuration

Returns: result containing minimum location and value

◆ LBFGS()

template<typename Function >

VectorResult MathOpt::LBFGS	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		int	theMemorySize = 10,
		const Config &	theConfig = Config() )

L-BFGS (Limited-memory BFGS) for large-scale optimization. Uses only the m most recent {s, y} pairs instead of full Hessian. Memory: O(m*n) instead of O(n^2).

Template Parameters

Function type with Value and Gradient methods

Parameters

theFunc	function object
theStartingPoint	initial guess
theMemorySize	number of gradient pairs to store (default 10)
theConfig	solver configuration

Returns: result containing minimum location and value

◆ MultiStart()

template<typename Function >

VectorResult MathOpt::MultiStart	(	Function &	theFunc,
		const math_Vector &	theLowerBounds,
		const math_Vector &	theUpperBounds,
		const GlobalConfig &	theConfig = GlobalConfig() )

Multi-start local optimization.

Runs multiple local optimizations from random starting points and returns the best result found. Uses Powell's method for local optimization (gradient-free).

Template Parameters

Function type with Value(const math_Vector&, double&) method

Parameters

theFunc	function to minimize
theLowerBounds	lower bounds
theUpperBounds	upper bounds
theConfig	optimization configuration

Returns: result containing best solution found

◆ Newton()

template<typename Function >

VectorResult MathOpt::Newton	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		const NewtonConfig &	theConfig = NewtonConfig() )

Newton's method for N-dimensional minimization using Hessian.

Fastest convergence near minimum (quadratic) but requires Hessian computation. Uses line search for global convergence and Hessian regularization when the Hessian is not positive definite.

Algorithm:

Compute gradient g and Hessian H at current point
If H is not positive definite, regularize: H = H + lambda*I
Solve H * p = -g for search direction p
Perform line search along p
Update x = x + alpha * p
Repeat until convergence

Template Parameters

Function

type with:

Value(const math_Vector&, double&) for function value
Gradient(const math_Vector&, math_Vector&) for gradient
Hessian(const math_Vector&, math_Matrix&) for Hessian

Parameters

theFunc	function object with value, gradient, and Hessian
theStartingPoint	initial guess
theConfig	solver configuration

Returns: result containing minimum location and value

◆ NewtonBounded()

template<typename Function >

VectorResult MathOpt::NewtonBounded	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		const math_Vector &	theLowerBounds,
		const math_Vector &	theUpperBounds,
		const NewtonConfig &	theConfig = NewtonConfig() )

Newton's method with bound constraints.

Minimizes f(x) subject to theLowerBounds <= x <= theUpperBounds. Uses projected gradient approach similar to BFGSBounded.

Template Parameters

Function type with Value, Gradient, and Hessian methods

Parameters

theFunc	function object
theStartingPoint	initial guess
theLowerBounds	lower bounds for each variable
theUpperBounds	upper bounds for each variable
theConfig	solver configuration

Returns: result containing minimum location and value

◆ NewtonModified()

template<typename Function >

VectorResult MathOpt::NewtonModified	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		const NewtonConfig &	theConfig = NewtonConfig() )

Modified Newton's method with automatic Hessian regularization. Adds diagonal elements to ensure positive definiteness using an adaptive regularization strategy.

Template Parameters

Function type with Value, Gradient, and Hessian methods

Parameters

theFunc	function object
theStartingPoint	initial guess
theConfig	solver configuration

Returns: result containing minimum location and value

◆ NewtonNumerical()

template<typename Function >

VectorResult MathOpt::NewtonNumerical	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		double	theGradStep = 1.0e-8,
		double	theHessStep = 1.0e-6,
		const NewtonConfig &	theConfig = NewtonConfig() )

Newton's method with fully numerical derivatives. Computes both gradient and Hessian using finite differences.

Template Parameters

Function type with Value(const math_Vector&, double&) method only

Parameters

theFunc	function object
theStartingPoint	initial guess
theGradStep	step size for numerical gradient
theHessStep	step size for numerical Hessian
theConfig	solver configuration

Returns: result containing minimum location and value

◆ NewtonNumericalHessian()

template<typename Function >

VectorResult MathOpt::NewtonNumericalHessian	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		double	theHessStep = 1.0e-6,
		const NewtonConfig &	theConfig = NewtonConfig() )

Newton's method with numerical Hessian. Computes Hessian using finite differences when analytical Hessian is not available.

Template Parameters

Function

type with:

Value(const math_Vector&, double&) for function value
Gradient(const math_Vector&, math_Vector&) for gradient

Parameters

theFunc	function object
theStartingPoint	initial guess
theHessStep	step size for numerical Hessian
theConfig	solver configuration

Returns: result containing minimum location and value

◆ PolishCoordinateWise()

template<typename Function >

void MathOpt::PolishCoordinateWise	(	Function &	theFunc,
		math_Vector &	thePosition,
		double &	theValue,
		const math_Vector &	theLowerBounds,
		const math_Vector &	theUpperBounds,
		double	theTolerance,
		int	theMaxPolishEvals,
		int &	theEvalCount )

Coordinate-wise polishing using Brent's 1D minimization. For each dimension, performs Brent's method directly on the single coordinate, modifying only one element of the position vector per evaluation (zero-allocation).

Template Parameters

Function type with Value(const math_Vector&, double&) method

Parameters

theFunc	function to minimize
thePosition	current best position (updated in place)
theValue	current best value (updated in place)
theLowerBounds	lower bounds for each variable
theUpperBounds	upper bounds for each variable
theTolerance	convergence tolerance for Brent's method
theMaxPolishEvals	maximum total function evaluations for polishing
theEvalCount	output: number of function evaluations used

◆ Powell()

template<typename Function >

VectorResult MathOpt::Powell	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		const Config &	theConfig = Config() )

Powell's conjugate direction method for N-dimensional minimization. A gradient-free optimization algorithm that uses conjugate directions.

Algorithm:

Start with N linearly independent directions (coordinate axes)
Perform line minimization along each direction
Replace one direction with the overall displacement direction
Repeat until convergence

Advantages:

No gradient required
Generates conjugate directions for quadratic functions
Robust for non-smooth functions

Disadvantages:

Slower than gradient methods for smooth functions
May lose direction independence over iterations

Template Parameters

Function type with Value(const math_Vector&, double&) method

Parameters

theFunc	function to minimize
theStartingPoint	initial guess (N-dimensional)
theConfig	solver configuration

Returns: result containing minimum location and value

◆ PowellWithDirections()

template<typename Function >

VectorResult MathOpt::PowellWithDirections	(	Function &	theFunc,
		const math_Vector &	theStartingPoint,
		const math_Matrix &	theInitialDirections,
		const Config &	theConfig = Config() )

Powell's method with custom initial directions. Allows specifying the initial direction set instead of coordinate axes.

Template Parameters

Function type with Value(const math_Vector&, double&) method

Parameters

theFunc	function to minimize
theStartingPoint	initial guess
theInitialDirections	initial direction set (N x N matrix, directions as rows)
theConfig	solver configuration

Returns: result containing minimum location and value

◆ PSO() [1/2]

template<typename Function >

VectorResult MathOpt::PSO	(	Function &	theFunc,
		const math_Vector &	theLowerBounds,
		const math_Vector &	theUpperBounds,
		const PSOConfig &	theConfig,
		const NCollection_DynamicArray< PSOSeedParticle > *	theSeeds,
		PSOStats *	theStats = nullptr )

Particle Swarm Optimization for global minimization.

PSO is a stochastic, population-based optimization algorithm inspired by social behavior of bird flocking or fish schooling.

Algorithm:

Initialize particles with random positions and velocities
Evaluate fitness (function value) for each particle
Update each particle's personal best if current position is better
Update global best across all particles
Update velocities: v = omega*v + c1*r1*(pbest-x) + c2*r2*(gbest-x)
Update positions: x = x + v
Repeat until convergence or max iterations

Properties:

Gradient-free: does not require derivatives
Global: can escape local minima
Stochastic: results depend on random initialization
Bound-constrained: handles box constraints naturally

Template Parameters

Function type with Value(const math_Vector&, double&) method

Parameters

theFunc	function to minimize
theLowerBounds	lower bounds for each variable
theUpperBounds	upper bounds for each variable
theConfig	PSO configuration
theSeeds	optional seed particles for initialization
theStats	optional output statistics

Returns: result containing best solution found

◆ PSO() [2/2]

template<typename Function >

VectorResult MathOpt::PSO	(	Function &	theFunc,
		const math_Vector &	theLowerBounds,
		const math_Vector &	theUpperBounds,
		const PSOConfig &	theConfig = PSOConfig() )

Particle Swarm Optimization for global minimization (basic overload).

Template Parameters

Function type with Value(const math_Vector&, double&) method

Parameters

theFunc	function to minimize
theLowerBounds	lower bounds for each variable
theUpperBounds	upper bounds for each variable
theConfig	PSO configuration

Returns: result containing best solution found

◆ Uzawa()

UzawaResult MathOpt::Uzawa	(	const math_Matrix &	theCont,
		const math_Vector &	theSecont,
		const math_Vector &	theStartingPoint,
		int	theNce,
		int	theNci,
		const UzawaConfig &	theConfig = UzawaConfig() )

inline

Solve constrained least squares using Uzawa algorithm.

Solves: min ||X - X0||^2 subject to C*X = S

For equality constraints only, uses direct Crout decomposition. For mixed equality/inequality constraints, uses iterative Uzawa method.

The Uzawa algorithm is a dual decomposition method that:

Updates primal variables X to minimize Lagrangian
Updates dual variables (Lagrange multipliers) for constraint violations

Parameters

theCont	constraint matrix C (Nce+Nci rows x N cols)
theSecont	right-hand side S
theStartingPoint	initial point X0
theNce	number of equality constraints (first rows)
theNci	number of inequality constraints (last rows, C*X <= S)
theConfig	algorithm configuration

Returns: UzawaResult with solution and auxiliary data

◆ UzawaEquality()

UzawaResult MathOpt::UzawaEquality	(	const math_Matrix &	theCont,
		const math_Vector &	theSecont,
		const math_Vector &	theStartingPoint,
		const UzawaConfig &	theConfig = UzawaConfig() )

inline

Solve constrained least squares with equality constraints only.

Convenience function for C*X = S with min ||X - X0||.

Parameters

theCont	constraint matrix C
theSecont	right-hand side S
theStartingPoint	initial point X0
theConfig	algorithm configuration

Returns: UzawaResult with solution

Data Structures

Enumerations

Functions

Detailed Description

Enumeration Type Documentation

◆ ConjugateGradientFormula

◆ GlobalStrategy

◆ PSOBoundaryMode

◆ PSOInertiaSchedule

◆ PSOInitMode

Function Documentation

◆ BFGS()

◆ BFGSBounded()

◆ BFGSNumerical()

◆ Brent()

◆ BrentWithBracket()

◆ DifferentialEvolution()

◆ FRPR()

◆ FRPRNumerical()

◆ GlobalMinimum() [1/2]

◆ GlobalMinimum() [2/2]

◆ Golden()

◆ LBFGS()

◆ MultiStart()

◆ Newton()

◆ NewtonBounded()

◆ NewtonModified()

◆ NewtonNumerical()

◆ NewtonNumericalHessian()

◆ PolishCoordinateWise()

◆ Powell()

◆ PowellWithDirections()

◆ PSO() [1/2]

◆ PSO() [2/2]

◆ Uzawa()

◆ UzawaEquality()