1. Affine convex functions

In optimization problems [16, 17, 19, 23–27], one can use an affine manifold as a pair ð Þ M; Γ , where M is a smooth real n-dimensional manifold, and Γ is an affine symmetric connection on M. The connection Γ produces auto-parallel curves x tð Þ via ODE system

as a PDEs system (a particular case of a Frobenius-Mayer system of PDEs) with <sup>1</sup>

<sup>∂</sup>xk <sup>¼</sup> <sup>∂</sup>Γ<sup>h</sup>

ikj <sup>¼</sup> <sup>∂</sup>Γ<sup>h</sup> ij <sup>∂</sup>xk � <sup>∂</sup>Γ<sup>h</sup> ki <sup>∂</sup>xj <sup>þ</sup> <sup>Γ</sup><sup>l</sup>

Of course this only means the curvature tensor is zero on the topologically trivial region we

topologically trivial regions, so this allows us to deduce that the curvature tensor vanishes

ð Þ<sup>x</sup> would now be everywhere nonzero co-vector fields; but there are topologies, for example, S<sup>2</sup>

which we know such things do not exist. Therefore, there are topological manifolds for which we are

The following theorem is well-known [16, 17, 19, 23]. Due to its importance, now we offer new proofs (based on catastrophe theory, decomposing a tensor into a specific product, and using

(1) If f is regular or has only one minimum point, then there exists a connection Γ such that f is affine

Proof. For the Hessian ð Þ HessΓf ij be positive semidefinite, we need n conditions like inequalities

ij is <sup>1</sup>

(2) If f has a maximum point x0, then there is no connection Γ making f affine convex throughout.

∂3 f

∂xi ∂xj

∂3 f

ikj <sup>¼</sup> <sup>0</sup>, Rh

ikj ¼ 0.

∂xi ∂xj

∂f <sup>∂</sup>xh <sup>R</sup><sup>h</sup>

Corollary 1.1 If there exists n linear affine functions f <sup>l</sup>

Remark 1.1 There is actually no need to extend df <sup>l</sup>

forced to work on topologically trivial regions.

Theorem 1.3 Let f : <sup>M</sup> ! <sup>R</sup> be a C<sup>2</sup> function.

and equalities. The number of unknowns Γ<sup>h</sup>

equalities using slackness variables.

independent, then Γ is flat, that is, Rh

used to set up our co-vector fields df <sup>l</sup>

throughout the manifold.

slackness variables).

Since,

it follows

df l

convex.

equations and the unknown function f , then we need the complete integrability conditions

<sup>∂</sup>xk <sup>¼</sup> <sup>∂</sup><sup>3</sup>

ij <sup>∂</sup>xk <sup>þ</sup> <sup>Γ</sup><sup>l</sup>

f ∂xk∂xi

ijΓh kl ! <sup>∂</sup><sup>f</sup>

<sup>∂</sup>xh ,

ijΓh kl � <sup>Γ</sup><sup>l</sup> kiΓ<sup>h</sup> jl:

, l ¼ 1, …, n on Mð Þ ; Γ , whose df <sup>l</sup> are linearly

Bilevel Disjunctive Optimization on Affine Manifolds http://dx.doi.org/10.5772/intechopen.75643

ð Þx to the entire manifold. If this could be done, then

<sup>2</sup> <sup>n</sup><sup>2</sup>ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> : The inequalities can be replaced by

ð Þx . But we can always cover any manifold by an atlas of

∂xj :

<sup>2</sup> n nð Þ þ 1

117

, for

$$
\ddot{\boldsymbol{x}}^h(t) + \Gamma^h\_{\dot{\boldsymbol{y}}}(\boldsymbol{x}(t))\dot{\boldsymbol{x}}^i(t)\dot{\boldsymbol{x}}^j(t) = \boldsymbol{0}.
$$

They are used for defining the convexity of subsets in M and convexity of functions f : D ⊂ M ! R (see also [3, 6]).

Definition 1.1 An affine manifold Mð Þ ; Γ is called autoparallely complete if any auto-parallel x tð Þ starting at p∈ M is defined for all values of the parameter t∈ R.

Theorem 1.1 [1] Let M be a (Hausdorff, connected, smooth) compact n-manifold endowed with an affine connection Γ and let p∈ M. If the holonomy group Holpð Þ Γ (regarded as a subgroup of the group Gl TpM of all the linear automorphisms of the tangent space TpM) has compact closure, then Mð Þ ; <sup>Γ</sup> is autoparallely complete.

Let ð Þ <sup>M</sup>; <sup>Γ</sup> be an auto-parallely complete affine manifold. For a <sup>C</sup><sup>2</sup> function <sup>f</sup> : <sup>M</sup> ! <sup>R</sup>, we define the tensor HessΓf of components

$$(H \text{ess}\_{\Gamma} f)\_{\vec{\eta}} = \frac{\partial^2 f}{\partial \mathbf{x}^i \partial \mathbf{x}^j} - \Gamma^h\_{\vec{\eta}} \frac{\partial f}{\partial \mathbf{x}^h} \dots$$

Definition 1.2 A C<sup>2</sup> function f : <sup>M</sup> ! <sup>R</sup> is called:

(1) linear affine with respect to Γ if HessΓf ¼ 0, throughout;

(2) affine convex (convex with respect to Γ) if Hess<sup>Γ</sup> f ≽0 (positive semidefinite), throughout.

The function f is: (1) linear affine if its restriction f xt ð Þ ð Þ on each autoparallel x tð Þ satisfies f xt ð Þ¼ ð Þ at þ b, for some numbers a, b that may depend on x tð Þ; (2) affine convex if its restriction f xt ð Þ ð Þ is convex on each auto-parallel x tð Þ.

Theorem 1.2 If there exists a linear affine nonconstant function f on Mð Þ ; Γ , then the curvature tensor field Rh ikj is in Ker df .

Proof. For given Γ, if we consider

$$\frac{\partial^2 f}{\partial x^i \partial x^j} = \Gamma^h\_{\,\,ij} \frac{\partial f}{\partial x^h}$$

as a PDEs system (a particular case of a Frobenius-Mayer system of PDEs) with <sup>1</sup> <sup>2</sup> n nð Þ þ 1 equations and the unknown function f , then we need the complete integrability conditions

$$
\frac{\partial^3 f}{\partial x^i \partial x^j \partial x^k} = \frac{\partial^3 f}{\partial x^k \partial x^i \partial x^j} \dots
$$

Since,

1. Affine convex functions

116 Optimization Algorithms - Examples

M ! R (see also [3, 6]).

autoparallely complete.

field Rh

ikj is in Ker df .

Proof. For given Γ, if we consider

define the tensor HessΓf of components

Definition 1.2 A C<sup>2</sup> function f : <sup>M</sup> ! <sup>R</sup> is called:

tion f xt ð Þ ð Þ is convex on each auto-parallel x tð Þ.

(1) linear affine with respect to Γ if HessΓf ¼ 0, throughout;

In optimization problems [16, 17, 19, 23–27], one can use an affine manifold as a pair ð Þ M; Γ , where M is a smooth real n-dimensional manifold, and Γ is an affine symmetric connection on

> ijð Þ x tð Þ x\_ i ð Þt x\_ j ðÞ¼ t 0:

They are used for defining the convexity of subsets in M and convexity of functions f : D ⊂

Definition 1.1 An affine manifold Mð Þ ; Γ is called autoparallely complete if any auto-parallel x tð Þ

Theorem 1.1 [1] Let M be a (Hausdorff, connected, smooth) compact n-manifold endowed with an affine connection Γ and let p∈ M. If the holonomy group Holpð Þ Γ (regarded as a subgroup of the group Gl TpM of all the linear automorphisms of the tangent space TpM) has compact closure, then Mð Þ ; <sup>Γ</sup> is

Let ð Þ <sup>M</sup>; <sup>Γ</sup> be an auto-parallely complete affine manifold. For a <sup>C</sup><sup>2</sup> function <sup>f</sup> : <sup>M</sup> ! <sup>R</sup>, we

f ∂xi

<sup>∂</sup>xj � <sup>Γ</sup><sup>h</sup> ij ∂f <sup>∂</sup>xh :

ð Þ HessΓ<sup>f</sup> ij <sup>¼</sup> <sup>∂</sup><sup>2</sup>

(2) affine convex (convex with respect to Γ) if Hess<sup>Γ</sup> f ≽0 (positive semidefinite), throughout.

∂2 f ∂xi

The function f is: (1) linear affine if its restriction f xt ð Þ ð Þ on each autoparallel x tð Þ satisfies f xt ð Þ¼ ð Þ at þ b, for some numbers a, b that may depend on x tð Þ; (2) affine convex if its restric-

Theorem 1.2 If there exists a linear affine nonconstant function f on Mð Þ ; Γ , then the curvature tensor

<sup>∂</sup>xj <sup>¼</sup> <sup>Γ</sup><sup>h</sup> ij ∂f ∂xh

M. The connection Γ produces auto-parallel curves x tð Þ via ODE system

ð Þþ <sup>t</sup> <sup>Γ</sup><sup>h</sup>

x€h

starting at p∈ M is defined for all values of the parameter t∈ R.

$$\frac{\partial^3 f}{\partial \alpha^i \partial \alpha^j \partial \alpha^k} = \left( \frac{\partial \Gamma^h\_{\vec{ij}}}{\partial \alpha^k} + \Gamma^l\_{\vec{ij}} \Gamma^h\_{kl} \right) \frac{\partial f}{\partial \alpha^{li}} \dots$$

it follows

$$\frac{\partial f}{\partial \mathbf{x}^h} \mathbf{R}^h{}\_{ik\dot{j}} = \mathbf{0},\\ \mathbf{R}^h{}\_{ik\dot{j}} = \frac{\partial \Gamma^h\_{\dot{i}\dot{j}}}{\partial \mathbf{x}^k} - \frac{\partial \Gamma^h\_{k\dot{i}}}{\partial \mathbf{x}^{\dot{j}}} + \Gamma^l\_{\dot{i}\dot{j}} \Gamma^h\_{kl} - \Gamma^l\_{ki} \Gamma^h\_{\dot{j}l}.$$

Corollary 1.1 If there exists n linear affine functions f <sup>l</sup> , l ¼ 1, …, n on Mð Þ ; Γ , whose df <sup>l</sup> are linearly independent, then Γ is flat, that is, Rh ikj ¼ 0.

Of course this only means the curvature tensor is zero on the topologically trivial region we used to set up our co-vector fields df <sup>l</sup> ð Þx . But we can always cover any manifold by an atlas of topologically trivial regions, so this allows us to deduce that the curvature tensor vanishes throughout the manifold.

Remark 1.1 There is actually no need to extend df <sup>l</sup> ð Þx to the entire manifold. If this could be done, then df l ð Þ<sup>x</sup> would now be everywhere nonzero co-vector fields; but there are topologies, for example, S<sup>2</sup> , for which we know such things do not exist. Therefore, there are topological manifolds for which we are forced to work on topologically trivial regions.

The following theorem is well-known [16, 17, 19, 23]. Due to its importance, now we offer new proofs (based on catastrophe theory, decomposing a tensor into a specific product, and using slackness variables).

Theorem 1.3 Let f : <sup>M</sup> ! <sup>R</sup> be a C<sup>2</sup> function.

(1) If f is regular or has only one minimum point, then there exists a connection Γ such that f is affine convex.

(2) If f has a maximum point x0, then there is no connection Γ making f affine convex throughout.

Proof. For the Hessian ð Þ HessΓf ij be positive semidefinite, we need n conditions like inequalities and equalities. The number of unknowns Γ<sup>h</sup> ij is <sup>1</sup> <sup>2</sup> <sup>n</sup><sup>2</sup>ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> : The inequalities can be replaced by equalities using slackness variables.

The first central idea for the proof is to use the catastrophe theory, since almost all families f xð Þ ; <sup>c</sup> , <sup>x</sup> <sup>¼</sup> <sup>x</sup><sup>1</sup>;…; ; xn <sup>∈</sup> <sup>R</sup><sup>n</sup>, <sup>c</sup> <sup>¼</sup> ð Þ <sup>c</sup>1;…; cm <sup>∈</sup> <sup>R</sup><sup>m</sup>, of real differentiable functions, with <sup>m</sup> <sup>≤</sup> <sup>4</sup> parameters, are structurally stable and are equivalent, in the vicinity of any point, with one of the following forms [15]:

To contradict, we fix an auto-parallel γð Þt , t∈ ½ Þ 0; e , starting from minimum point x<sup>0</sup> ¼ γð Þ0 ,

But this result depends on the direction v (different values along two different auto-parallels). In some particular cases, we can eliminate the dependence on the vector v. For example, the

ijð Þ x<sup>0</sup>

ijð Þ x<sup>0</sup>

,

∂2 f <sup>∂</sup>xh∂xl ð Þ� <sup>x</sup><sup>0</sup>

> ∂2 f <sup>∂</sup>xh∂xk ð Þ� <sup>x</sup><sup>0</sup>

> > ð Þ x<sup>0</sup>

∂2 f <sup>∂</sup>xh∂xk ð Þ <sup>x</sup><sup>0</sup> 

∂σij

<sup>∂</sup>xk ð Þ¼ <sup>x</sup><sup>0</sup> <sup>0</sup>:

<sup>∂</sup><sup>y</sup> <sup>¼</sup> <sup>3</sup>y<sup>2</sup> <sup>þ</sup> <sup>3</sup> and f has no critical

<sup>21</sup> ¼ 0, h ¼ 1, 2,

<sup>∂</sup><sup>y</sup> <sup>¼</sup> <sup>2</sup>ye� <sup>x</sup>2þy<sup>2</sup> ð Þ and f has a unique critical

<sup>∂</sup>xk ð Þ� <sup>x</sup><sup>0</sup> <sup>Γ</sup><sup>h</sup>

ijð Þ x<sup>0</sup>

ahð Þ x<sup>0</sup>

∂2f <sup>∂</sup>xh∂xk ð Þ� x<sup>0</sup>

> ∂σij <sup>∂</sup>xl ð Þ <sup>x</sup><sup>0</sup>

> > ∂σij <sup>∂</sup>xk ð Þ <sup>x</sup><sup>0</sup>

vk

∂2 f <sup>∂</sup>xh∂xk ð Þ x<sup>0</sup> vk ∂σij <sup>∂</sup>xk ð Þ x<sup>0</sup>

Bilevel Disjunctive Optimization on Affine Manifolds http://dx.doi.org/10.5772/intechopen.75643

:

119

∂3f ∂xi ∂xj

tangent to γ\_ð Þ¼ 0 v and we compute (via l'Hôpital rule)

bijð Þ¼ γð Þt

∂3 f

> ∂3 f

<sup>∂</sup>xl ð Þ� <sup>x</sup><sup>0</sup> <sup>Γ</sup><sup>h</sup>

<sup>∂</sup>xk ð Þ� <sup>x</sup><sup>0</sup> <sup>Γ</sup><sup>h</sup>

<sup>∂</sup>xh∂xl ð Þ¼ <sup>x</sup><sup>0</sup> <sup>r</sup> ah

ijð Þ x<sup>0</sup>

∂2 f <sup>∂</sup>xh∂xk ð Þ� <sup>x</sup><sup>0</sup>

In this particular condition, we can show that we can build connections of previous type good

Example 1.1 (for the first part of the theorem) Let us consider the function f : <sup>R</sup><sup>2</sup> ! <sup>R</sup>,f xð Þ¼ ; <sup>y</sup>

point. Moreover, the Euclidean Hessian of f is not positive semi-definite overall. Let us make the above

Example 1.2 (for one minimum point) Let us consider the function f : <sup>R</sup><sup>2</sup> ! <sup>R</sup>,f xð Þ¼ ; <sup>y</sup> <sup>1</sup> � <sup>e</sup>�

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>2</sup>xe� <sup>x</sup>2þy<sup>2</sup> ð Þ, <sup>∂</sup><sup>f</sup>

<sup>22</sup> <sup>¼</sup> <sup>6</sup><sup>y</sup> � <sup>1</sup> 3x<sup>2</sup> þ 3y<sup>2</sup> þ 2

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>3</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup>, <sup>∂</sup><sup>f</sup>

, Γ h <sup>12</sup> ¼ Γ h

∂2 f

<sup>∂</sup>xk ð Þ� <sup>x</sup><sup>0</sup> <sup>Γ</sup><sup>h</sup>

Let us lightning our previous statements by the following examples.

ij <sup>¼</sup> <sup>0</sup>, i, j, h <sup>¼</sup> <sup>1</sup>, <sup>2</sup>. Then <sup>∂</sup><sup>f</sup>

construction for <sup>σ</sup>ijð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>δ</sup>ij. Taking a<sup>1</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>, we obtain the connection

, Γ h

∂xi ∂xj

> ∂xi ∂xj

ah ð Þ x<sup>0</sup>

∂3 f

¼ r

A particular condition for independence on v is

∂xi ∂xj

t!0

bijð Þ¼ x0; v lim

conditions

are sufficient to do this.

1.1. Lightning through examples

<sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>y</sup><sup>3</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup> <sup>þ</sup> <sup>3</sup>y and <sup>Γ</sup><sup>h</sup>

that is not unique.

<sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> and <sup>Γ</sup><sup>h</sup>

Γ h

<sup>11</sup> <sup>¼</sup> <sup>6</sup><sup>x</sup> � <sup>1</sup> 3x<sup>2</sup> þ 3y<sup>2</sup> þ 2

ij <sup>¼</sup> <sup>0</sup>, i, j, h <sup>¼</sup> <sup>1</sup>, <sup>2</sup>. Then <sup>∂</sup><sup>f</sup>

everywhere.

We eliminate the case with maximum point, that is., Morse 0-saddle and the saddle point. Around each critical point (in a chart), the canonical form f xð Þ ; c is affine convex, with respect to appropriate locally defined linear connections that can be found easily. Using change of coordinates and the partition of unity, we glue all these connections to a global one, making f xð Þ ; c affine convex on M.

At any critical point <sup>x</sup>0, the affine Hessian HessΓ<sup>f</sup> is reduced to Euclidean Hessian, <sup>∂</sup>2<sup>f</sup> ∂xi <sup>∂</sup>xj ð Þ x<sup>0</sup> . Then the maximum point condition or the saddle condition is contradictory to affine convexity condition.

A direct proof based on decomposition of a tensor: Let ð Þ M; Γ be an affine manifold and <sup>f</sup> : <sup>M</sup> ! <sup>R</sup> be a <sup>C</sup><sup>2</sup> function.

Suppose f has no critical points (is regular). If the function f is not convex with respect to Γ, we look to find a new connection Γ h ij <sup>¼</sup> <sup>Γ</sup><sup>h</sup> ij <sup>þ</sup> <sup>T</sup><sup>h</sup> ij, with the unknown a tensor field T<sup>h</sup> ij, such that

$$\frac{\partial^2 f}{\partial \mathbf{x}^i \partial \mathbf{x}^j}(\mathbf{x}) - \overline{\Gamma}\_{ij}^h(\mathbf{x}) \frac{\partial f}{\partial \mathbf{x}^h}(\mathbf{x}) = \sigma\_{i\bar{j}}(\mathbf{x}), \mathbf{x} \in M\_{\mathbf{x}^h}$$

where σijð Þx is a positive semi-definite tensor. A very particular solution is the decomposition Th ijð Þ¼ <sup>x</sup> ahð Þ<sup>x</sup> bijð Þ<sup>x</sup> , where the vector field <sup>a</sup> has the property

$$D\_q f = d^h(\mathfrak{x}) \frac{\partial f}{\partial \mathfrak{x}^h}(\mathfrak{x}) \neq 0, \mathfrak{x} \in M$$

and the tensor bij is

$$b\_{\overrightarrow{\eta}}(\mathbf{x}) = \frac{1}{D\_{\overrightarrow{\eta}}f} \left( \frac{\partial^2 f}{\partial \mathbf{x}^i \partial \mathbf{x}^j}(\mathbf{x}) - \Gamma^h\_{\overrightarrow{\eta}}(\mathbf{x}) \, \frac{\partial f}{\partial \mathbf{x}^h}(\mathbf{x}) - \sigma\_{\overrightarrow{\eta}}(\mathbf{x}) \right), \mathbf{x} \in M.$$

Remark 1.2 The connection Γ h ij is strongly dependent on both the function f and the tensor field σij.

Suppose f has a minimum point x0. In this case, observe that we must have the condition σijð Þ¼ x<sup>0</sup> ∂2 f ∂xi <sup>∂</sup>xj ð Þ x<sup>0</sup> . Can we make the previous reason for x 6¼ x<sup>0</sup> and then extend the obtained connection by continuity? The answer is generally negative. Indeed, let us compute

$$b\_{\vec{\eta}}(\mathbf{x}\_0) = \lim\_{\mathbf{x} \to \mathbf{x}\_0} \frac{1}{D\_{\mathbf{q}}f} \left( \frac{\partial^2 f}{\partial \mathbf{x}^i \partial \mathbf{x}^j}(\mathbf{x}) - \Gamma^h\_{\vec{\eta}}(\mathbf{x}) \frac{\partial f}{\partial \mathbf{x}^h}(\mathbf{x}) - \sigma\_{\vec{\eta}}(\mathbf{x}) \right).$$

Here we cannot plug in the point x<sup>0</sup> because we get <sup>0</sup> <sup>0</sup> , an indeterminate form. To contradict, we fix an auto-parallel γð Þt , t∈ ½ Þ 0; e , starting from minimum point x<sup>0</sup> ¼ γð Þ0 , tangent to γ\_ð Þ¼ 0 v and we compute (via l'Hôpital rule)

$$b\_{\vec{\eta}}(\mathbf{x}\_{0};\boldsymbol{\upsilon}) = \lim\_{t \to 0} b\_{\vec{\eta}}(\boldsymbol{\upsilon}(t)) = \frac{\left(\frac{\partial^{\boldsymbol{\beta}}f}{\partial \mathbf{x}^{\boldsymbol{\prime}} \partial \mathbf{x}^{\boldsymbol{\prime}} \partial \mathbf{x}^{\boldsymbol{\prime}}}(\mathbf{x}\_{0}) - \boldsymbol{\Gamma}^{h}\_{\vec{\eta}}(\mathbf{x}\_{0}) \frac{\partial^{2}f}{\partial \mathbf{x}^{h} \partial \mathbf{x}^{\boldsymbol{\prime}}}(\mathbf{x}\_{0}) - \frac{\partial \boldsymbol{\varepsilon}\_{\vec{\eta}}}{\partial \mathbf{x}^{\boldsymbol{\prime}}}(\mathbf{x}\_{0})\right) \mathbf{z}^{k}}{a^{\boldsymbol{\prime}}(\mathbf{x}\_{0}) \frac{\partial^{2}f}{\partial \mathbf{x}^{\boldsymbol{\prime}} \partial \mathbf{x}^{\boldsymbol{\prime}}}(\mathbf{x}\_{0}) \mathbf{z}^{k}}.$$

But this result depends on the direction v (different values along two different auto-parallels).

In some particular cases, we can eliminate the dependence on the vector v. For example, the conditions

$$\frac{\partial^3 f}{\partial \mathbf{x}^i \partial \mathbf{x}^j \partial \mathbf{x}^l}(\mathbf{x}\_0) - \Gamma^l\_{\, \dot{\eta}}(\mathbf{x}\_0) \frac{\partial^2 f}{\partial \mathbf{x}^h \partial \mathbf{x}^l}(\mathbf{x}\_0) - \frac{\partial \sigma\_{\dot{\eta}}}{\partial \mathbf{x}^l}(\mathbf{x}\_0)$$

$$= \rho \left( \frac{\partial^3 f}{\partial \mathbf{x}^i \partial \mathbf{x}^j \partial \mathbf{x}^k}(\mathbf{x}\_0) - \Gamma^h\_{\, \dot{\eta}}(\mathbf{x}\_0) \frac{\partial^2 f}{\partial \mathbf{x}^h \partial \mathbf{x}^k}(\mathbf{x}\_0) - \frac{\partial \sigma\_{\dot{\eta}}}{\partial \mathbf{x}^k}(\mathbf{x}\_0) \right),$$

$$a^h(\mathbf{x}\_0) \frac{\partial^2 f}{\partial \mathbf{x}^h \partial \mathbf{x}^l}(\mathbf{x}\_0) = \rho \left( a^h(\mathbf{x}\_0) \frac{\partial^2 f}{\partial \mathbf{x}^h \partial \mathbf{x}^k}(\mathbf{x}\_0) \right)$$

are sufficient to do this.

The first central idea for the proof is to use the catastrophe theory, since almost all families f xð Þ ; <sup>c</sup> , <sup>x</sup> <sup>¼</sup> <sup>x</sup><sup>1</sup>;…; ; xn <sup>∈</sup> <sup>R</sup><sup>n</sup>, <sup>c</sup> <sup>¼</sup> ð Þ <sup>c</sup>1;…; cm <sup>∈</sup> <sup>R</sup><sup>m</sup>, of real differentiable functions, with <sup>m</sup> <sup>≤</sup> <sup>4</sup> parameters, are structurally stable and are equivalent, in the vicinity of any point, with one of

We eliminate the case with maximum point, that is., Morse 0-saddle and the saddle point. Around each critical point (in a chart), the canonical form f xð Þ ; c is affine convex, with respect to appropriate locally defined linear connections that can be found easily. Using change of coordinates and the partition of unity, we glue all these connections to a global one, making

At any critical point <sup>x</sup>0, the affine Hessian HessΓ<sup>f</sup> is reduced to Euclidean Hessian, <sup>∂</sup>2<sup>f</sup>

Then the maximum point condition or the saddle condition is contradictory to affine convexity

A direct proof based on decomposition of a tensor: Let ð Þ M; Γ be an affine manifold and

Suppose f has no critical points (is regular). If the function f is not convex with respect to Γ, we

where σijð Þx is a positive semi-definite tensor. A very particular solution is the decomposition

ijð Þx ∂f

Suppose f has a minimum point x0. In this case, observe that we must have the condition

<sup>∂</sup>xj ð Þ� <sup>x</sup> <sup>Γ</sup><sup>h</sup>

<sup>∂</sup>xj ð Þ x<sup>0</sup> . Can we make the previous reason for x 6¼ x<sup>0</sup> and then extend the obtained

ijð Þx ∂f

ij, with the unknown a tensor field T<sup>h</sup>

<sup>∂</sup>xh ð Þ¼ <sup>x</sup> <sup>σ</sup>ijð Þ<sup>x</sup> , x <sup>∈</sup> M,

<sup>∂</sup>xh ð Þ<sup>x</sup> 6¼ <sup>0</sup>, x <sup>∈</sup> <sup>M</sup>

<sup>∂</sup>xh ð Þ� <sup>x</sup> <sup>σ</sup>ijð Þ<sup>x</sup>

ij is strongly dependent on both the function f and the tensor field σij.

<sup>∂</sup>xh ð Þ� <sup>x</sup> <sup>σ</sup>ijð Þ<sup>x</sup>

<sup>0</sup> , an indeterminate form.

, x∈ M:

:

ij <sup>þ</sup> <sup>T</sup><sup>h</sup>

h ijð Þx ∂f

h ij <sup>¼</sup> <sup>Γ</sup><sup>h</sup>

<sup>∂</sup>xj ð Þ� <sup>x</sup> <sup>Γ</sup>

Daf <sup>¼</sup> ah

∂2 f ∂xi

> 1 Daf

ð Þx ∂f

<sup>∂</sup>xj ð Þ� <sup>x</sup> <sup>Γ</sup><sup>h</sup>

connection by continuity? The answer is generally negative. Indeed, let us compute

∂2 f ∂xi

∂2 f ∂xi

ijð Þ¼ <sup>x</sup> ahð Þ<sup>x</sup> bijð Þ<sup>x</sup> , where the vector field <sup>a</sup> has the property

1 Daf

h

bijð Þ¼ <sup>x</sup><sup>0</sup> lim<sup>x</sup>!x<sup>0</sup>

Here we cannot plug in the point x<sup>0</sup> because we get <sup>0</sup>

bijð Þ¼ x

∂xi <sup>∂</sup>xj ð Þ x<sup>0</sup> .

ij, such that

the following forms [15]:

118 Optimization Algorithms - Examples

f xð Þ ; c affine convex on M.

<sup>f</sup> : <sup>M</sup> ! <sup>R</sup> be a <sup>C</sup><sup>2</sup> function.

look to find a new connection Γ

condition.

Th

and the tensor bij is

σijð Þ¼ x<sup>0</sup>

Remark 1.2 The connection Γ

∂2 f ∂xi

A particular condition for independence on v is

$$\frac{\partial^3 f}{\partial \mathbf{x}^i \partial \mathbf{x}^j \partial \mathbf{x}^k}(\mathbf{x}\_0) - \Gamma^h\_{ij}(\mathbf{x}\_0) \frac{\partial^2 f}{\partial \mathbf{x}^h \partial \mathbf{x}^k}(\mathbf{x}\_0) - \frac{\partial \sigma\_{ij}}{\partial \mathbf{x}^k}(\mathbf{x}\_0) = \mathbf{0}.$$

In this particular condition, we can show that we can build connections of previous type good everywhere.

#### 1.1. Lightning through examples

Let us lightning our previous statements by the following examples.

Example 1.1 (for the first part of the theorem) Let us consider the function f : <sup>R</sup><sup>2</sup> ! <sup>R</sup>,f xð Þ¼ ; <sup>y</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>y</sup><sup>3</sup> <sup>þ</sup> <sup>3</sup><sup>x</sup> <sup>þ</sup> <sup>3</sup>y and <sup>Γ</sup><sup>h</sup> ij <sup>¼</sup> <sup>0</sup>, i, j, h <sup>¼</sup> <sup>1</sup>, <sup>2</sup>. Then <sup>∂</sup><sup>f</sup> <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>3</sup>x<sup>2</sup> <sup>þ</sup> <sup>3</sup>, <sup>∂</sup><sup>f</sup> <sup>∂</sup><sup>y</sup> <sup>¼</sup> <sup>3</sup>y<sup>2</sup> <sup>þ</sup> <sup>3</sup> and f has no critical point. Moreover, the Euclidean Hessian of f is not positive semi-definite overall. Let us make the above construction for <sup>σ</sup>ijð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>δ</sup>ij. Taking a<sup>1</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>, we obtain the connection

$$\overline{\Gamma}\_{11}^{\natural} = \frac{6\mathbf{x} - \mathbf{1}}{3\mathbf{x}^2 + 3\mathbf{y}^2 + 2}, \\ \overline{\Gamma}\_{22}^{\natural} = \frac{6\mathbf{y} - \mathbf{1}}{3\mathbf{x}^2 + 3\mathbf{y}^2 + 2}, \\ \overline{\Gamma}\_{12}^{\natural} = \overline{\Gamma}\_{21}^{\natural} = \mathbf{0}, \ \hbar = 1, 2, 3$$

that is not unique.

Example 1.2 (for one minimum point) Let us consider the function f : <sup>R</sup><sup>2</sup> ! <sup>R</sup>,f xð Þ¼ ; <sup>y</sup> <sup>1</sup> � <sup>e</sup>� <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> and <sup>Γ</sup><sup>h</sup> ij <sup>¼</sup> <sup>0</sup>, i, j, h <sup>¼</sup> <sup>1</sup>, <sup>2</sup>. Then <sup>∂</sup><sup>f</sup> <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>2</sup>xe� <sup>x</sup>2þy<sup>2</sup> ð Þ, <sup>∂</sup><sup>f</sup> <sup>∂</sup><sup>y</sup> <sup>¼</sup> <sup>2</sup>ye� <sup>x</sup>2þy<sup>2</sup> ð Þ and f has a unique critical minimum point ð Þ 0; 0 . However, the Euclidean Hessian of f is not positive semi-definite overall. We make previous reason for σij ¼ 2e � <sup>x</sup>2þy<sup>2</sup> ð Þδij, a<sup>1</sup> <sup>¼</sup> <sup>∂</sup><sup>f</sup> <sup>∂</sup><sup>x</sup> , a<sup>2</sup> <sup>¼</sup> <sup>∂</sup><sup>f</sup> ∂y . Hence we obtain Γ h ij <sup>¼</sup> <sup>T</sup><sup>h</sup> ij,

Our chapter is based also on some ideas in: [3] (convex mappings between Riemannian manifolds), [7] (geometric modeling in probability and statistics), [13] (arc length in metric and Finsler manifolds), [14] (applications of Hahn-Banach principle to moment and optimization problems), [21] (geodesic connectedness of semi-Riemannian manifolds), and [28] (tangent and cotangent bundles). For algorithms, we recommend the paper [20] (sequential and parallel

The auto-parallel curves x tð Þ on the affine manifold ð Þ M; Γ are solutions of the second order

x tð Þ¼ <sup>0</sup>; x0; ξ<sup>0</sup> x0, x t \_ð Þ¼ <sup>0</sup>; x0; ξ<sup>0</sup> ξ0:

Definition 2.1 Let D <sup>⊂</sup> M be open and connected and f : <sup>D</sup> ! <sup>R</sup> a C<sup>2</sup> function. The point x<sup>0</sup> <sup>∈</sup> D is called minimum (maximum) point of f conditioned by the auto-parallel system, together with initial conditions, if for the maximal solution x tð Þ ; x0; ξ<sup>0</sup> : I ! D, there exists a neighborhood It<sup>0</sup> of t<sup>0</sup> such that

f xt ð Þ ð Þ ; x0; ξ<sup>0</sup> ≥ ≤ð Þ f xð Þ<sup>0</sup> , ∀t∈ It<sup>0</sup> ⊂I:

Theorem 2.1 If x<sup>0</sup> ∈ D is an extremum point of f conditioned by the previous second order system,

Definition 2.2 The points x∈ D which are solutions of the equation df xð Þð Þ¼ ξ 0 are called critical

Theorem 2.2 If x<sup>0</sup> <sup>∈</sup> D is a conditioned critical point of the function f : <sup>D</sup> ! <sup>R</sup> of class C<sup>2</sup> constrained

f ∂xi

is strictly positive (negative), then x<sup>0</sup> is a minimum (maximum) point of f constrained by the auto-

<sup>∂</sup>xj � <sup>∂</sup><sup>f</sup> <sup>∂</sup>xh <sup>Γ</sup><sup>h</sup> ij ð Þ <sup>x</sup><sup>0</sup> <sup>ξ</sup><sup>i</sup>

0ξj 0

ðÞ¼ t 0,x tð Þ¼ <sup>0</sup> x0, x t \_ð Þ¼ <sup>0</sup> ξ0:

Bilevel Disjunctive Optimization on Affine Manifolds http://dx.doi.org/10.5772/intechopen.75643 121

2. Optimizations with autoparallel restrictions

x€h

ð Þþ <sup>t</sup> <sup>Γ</sup><sup>h</sup>

Obviously, the complete notation is x tð Þ ; x0; ξ<sup>0</sup> , with

ijð Þ x tð Þ x\_ i ð Þt x\_ j

algorithms).

2.1. Direct theory

then df xð Þ<sup>0</sup> ð Þ¼ ξ<sup>0</sup> 0.

parallel system.

points of f conditioned by the previous spray.

by the previous auto-parallel system and if the number

ð Þ Hessf ij <sup>ξ</sup><sup>i</sup>

0ξj

<sup>0</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup>

ODE system

$$
\overline{\Gamma}^1\_{11} = -\frac{2\mathbf{x}^3}{\mathbf{x}^2 + \mathbf{y}^2}, \\
\overline{\Gamma}^1\_{12} = \overline{\Gamma}^1\_{21} = -\frac{2\mathbf{x}^2 \mathbf{y}}{\mathbf{x}^2 + \mathbf{y}^2}, \\
\overline{\Gamma}^1\_{22} = -\frac{2\mathbf{x}\mathbf{y}^2}{\mathbf{x}^2 + \mathbf{y}^2}.
$$

$$
\overline{\Gamma}^2\_{11} = -\frac{2\mathbf{x}^2 \mathbf{y}}{\mathbf{x}^2 + \mathbf{y}^2}, \\
\overline{\Gamma}^2\_{12} = \overline{\Gamma}^2\_{21} = -\frac{2\mathbf{x}\mathbf{y}^2}{\mathbf{x}^2 + \mathbf{y}^2}, \\
\overline{\Gamma}^2\_{22} = -\frac{2\mathbf{y}^3}{\mathbf{x}^2 + \mathbf{y}^2}.
$$

Observe that limð Þ! <sup>x</sup>;<sup>y</sup> ð Þ <sup>0</sup>;<sup>0</sup> <sup>T</sup><sup>h</sup> ijð Þ¼ x; y 0. Hence take Γ h ijð Þ¼ 0; 0 0.

The next example shows what happens if we come out of the conditions of the previous theorem.

Example 1.3 Let us take the function f : <sup>R</sup> ! <sup>R</sup>,f xð Þ¼ <sup>x</sup><sup>3</sup>, where the critical point x <sup>¼</sup> <sup>0</sup> is an inflection point. We take <sup>Γ</sup>ð Þ¼� <sup>x</sup> <sup>1</sup> � <sup>2</sup> <sup>x</sup><sup>2</sup> , which is not defined at the critical point x ¼ 0, but the relation of convexity is realized by prolongation,

$$\sigma(\mathbf{x}) = f''(\mathbf{x}) - \Gamma(\mathbf{x})f'(\mathbf{x}) = \Im(\mathbf{x}^2 + 2\mathbf{x} + 2) > 0, \quad \forall \mathbf{x} \in \mathbb{R}.$$

Let us consider the ODE of auto-parallels

$$\mathbf{x''(t) - \left(1 + \frac{2}{t^2}\right)} \mathbf{x'(t)^2 = \mathbf{0}, t \neq \mathbf{0}.}$$

The solutions

$$\mathbf{x}(t) = -\frac{1}{2}\ln|-2+t^2-ct| + \frac{c}{\sqrt{8+c^2}}\text{ }\text{arctanh }\frac{2t-c}{\sqrt{8+c^2}} + c\_1$$

are auto-parallels on ð Þ R\f g 0; t1; t<sup>2</sup> ; Γ , where t1, t<sup>2</sup> are real solutions of �2 þ t <sup>2</sup> � ct <sup>¼</sup> <sup>0</sup>. These curves are extended at t ¼ 0 by continuity. The manifold ð Þ R; Γ is not auto-parallely complete. Since the image <sup>x</sup>ð Þ <sup>R</sup> is not a "segment", the function f : <sup>R</sup> ! <sup>R</sup>,f xð Þ¼ <sup>x</sup><sup>3</sup> is not globally convex.

Remark 1.3 For n <sup>≥</sup> <sup>2</sup>, there exists C<sup>1</sup> functions <sup>φ</sup> : <sup>R</sup><sup>n</sup> ! <sup>R</sup> which have two minimum points without having another extremum point. As example,

$$\varphi\left(\mathbf{x}^{1},\mathbf{x}^{2}\right) = \left(\mathbf{x}^{1^{2}} - 1\right)^{2} + \left(\mathbf{x}^{1^{2}}\mathbf{x}^{2} - \mathbf{x}^{1} - 1\right)^{2}$$

has two (global) minimum points p ¼ �ð Þ 1; 0 , q ¼ ð Þ 1; 2 .

The restriction

$$\varphi(\mathbf{x}^1, \mathbf{x}^2) = \left(\mathbf{x}^{1^4} + \mathbf{x}^{1^4}\mathbf{x}^{2^2} + 2\mathbf{x}^1 + 2\right) - \left(\mathbf{x}^{1^2} + 2\mathbf{x}^{1^3}\mathbf{x}^2 + 2\mathbf{x}^{1^2}\mathbf{x}^2\right), \mathbf{x}^1 > 0, \mathbf{x}^2 > 0$$

is difference of two affine convex functions (see Section 2).

Our chapter is based also on some ideas in: [3] (convex mappings between Riemannian manifolds), [7] (geometric modeling in probability and statistics), [13] (arc length in metric and Finsler manifolds), [14] (applications of Hahn-Banach principle to moment and optimization problems), [21] (geodesic connectedness of semi-Riemannian manifolds), and [28] (tangent and cotangent bundles). For algorithms, we recommend the paper [20] (sequential and parallel algorithms).
