Efficient solutions in V-KT-pseudoinvex multiobjective control problems: A characterization

Abstract

In this paper, we introduce a new condition on functionals involved in a multiobjective control problem, for which we define the V-KT-pseudoinvex control problem. We prove that a V-KT-pseudoinvex control problem is characterized so that a Kuhn–Tucker point is an efficient solution. We generalize recently obtained optimality results of known mathematical programming problems and control problems. We illustrate these results with an example.

Introduction

Control problems are often applied to engineering problems; for example, control design for autonomous vehicles [1] or impulsive control problems [2]. The search for solutions in scalar and multiobjective control problems has usually been carried out through the study of optimality conditions and of the functions that are involved. Recently, Arana et al. [3] have introduced KT-invexity conditions for the involved functionals of a scalar control problem, and proved that in order for Kuhn–Tucker points to be optimal solutions it is necessary and sufficient that the control problem is KT-invex, from works by Chandra et al. [4] and Mond and Smart [5]. In addition, Arana et al. [6] have extended the classes of functions introduced by Martin [7] to multiobjective functions in mathematical programming problems and have generalized his optimality results, providing characterization results for efficient solutions.

In this paper, our aim is to generalize these characterization results and classes of functions and functionals [3], [6] to multiobjective control problems. In this sense, we will introduce a new class of involved functionals for a multiobjective control problems and will prove that it is a necessary and sufficient condition for all Kuhn–Tucker points to be efficient solutions.

We consider the following multiobjective control problem mathematical formulation:

$$\begin{array}{cc}\hfill & \begin{array}{cc}(\text{MCP})\hfill & \text{Minimize}{\int }_a^{b}f(t\text{,}x\text{,}u)\mathit{dt}\hfill \\ \hfill & \text{subject to}:\hfill \end{array}\hfill \\ \hfill & x(a)=\alpha \text{,}x(b)=\beta \text{,}\hfill \\ \hfill & g(t\text{,}x\text{,}u)\leqq 0\text{,}t\in I\text{,}\hfill \\ \hfill & h(t\text{,}x\text{,}u)=\dot{x}\text{,}t\in I\text{.}\hfill \end{array}$$

Here $I=[a\text{,}b]$ is a real interval. Each $f=(f_1\text{,}{f}_2\text{,}\dots \text{,}{f}_p):I\times {\mathbb{R}}^n\times {\mathbb{R}}^m\to {\mathbb{R}}^p$, $g=(g_1\text{,}{g}_2\text{,}\dots \text{,}{g}_k):I\times {\mathbb{R}}^n\times {\mathbb{R}}^m\to {\mathbb{R}}^k$, $h=(h_1\text{,}{h}_2\text{,}\dots \text{,}{h}_n):I\times {\mathbb{R}}^n\times {\mathbb{R}}^m\to {\mathbb{R}}^n$ is a continuously differentiable function.

Denote the partial derivatives of f by $f_t$, $f_x$ and $f_u$, where

$$f_t=\left[\begin{array}{c}\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }t}\hfill \\ ⋮\hfill \\ \frac{\mathrm{\partial }{f}_p}{\mathrm{\partial }t}\hfill \end{array}\right]\text{,}{f}_x=\left[\begin{array}{cccc}\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }{x}^1}\hfill & \frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }{x}^2}\hfill & \dots \hfill & \frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }{x}^n}\hfill \\ ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \frac{\mathrm{\partial }{f}_p}{\mathrm{\partial }{x}^1}\hfill & \frac{\mathrm{\partial }{f}_p}{\mathrm{\partial }{x}^2}\hfill & \dots \hfill & \frac{\mathrm{\partial }{f}_p}{\mathrm{\partial }{x}^n}\hfill \end{array}\right]\text{,}{f}_u=\left[\begin{array}{cccc}\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }{u}^1}\hfill & \frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }{u}^2}\hfill & \dots \hfill & \frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }{u}^m}\hfill \\ ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \frac{\mathrm{\partial }{f}_p}{\mathrm{\partial }{u}^1}\hfill & \frac{\mathrm{\partial }{f}_p}{\mathrm{\partial }{u}^2}\hfill & \dots \hfill & \frac{\mathrm{\partial }{f}_p}{\mathrm{\partial }{u}^m}\hfill \end{array}\right]$$

using matrices with p rows, where the superscripts denote the vector components. Similarly, we have $g_t\text{,}{g}_x\text{,}{g}_u$ and $h_t\text{,}{h}_x\text{,}{h}_u$ using matrices with k and n rows, respectively.

Let X be the space of piecewise smooth state functions $x:I\to {\mathbb{R}}^n$, with derivative $\dot{x}$, and is equipped with the norm $\parallel x\parallel =\parallel x{\parallel }_{\infty }+\parallel \mathit{Dx}{\parallel }_{\infty }$, where the differentiation operator D is given by

$$w=\mathit{Dx}\iff x(t)=\alpha +{\int }_a^{t}w(s)\mathit{ds}\text{,}$$

and $\alpha$ is a given boundary value. Therefore, $D=d/\mathit{dt}$ except at discontinuities. Let Y be the space of piecewise continuous control functions $u:I\to {\mathbb{R}}^m$, with the uniform norm $\parallel ·{\parallel }_{\infty }$. For notational convenience $f(t\text{,}x(t)\text{,}u(t))$ will be written as $f(t\text{,}x\text{,}u)$, and similarly for g and h. We denote the set of feasible solutions of (MCP) by K, i. e.,

$$K=\{(x\text{,}u)\in X\times Y:x(a)=\alpha \text{,}x(b)=\beta \text{,}g(t\text{,}x(t)\text{,}\dot{x}(t))⩽0\text{,}h(t\text{,}x(t)\text{,}u(t))=\dot{x}(t)\text{,}t\in I\}$$

and define $F:X\times Y\to {\mathbb{R}}^p$ by $F(x\text{,}u)=(F_1(x\text{,}u)\text{,}\dots \text{,}{F}_p(x\text{,}u))={\int }_a^{b}f(t\text{,}x\text{,}u)\mathit{dt}$, $G:X\times Y\to {\mathbb{R}}^k$ by $G(x\text{,}u)=(G_1(x\text{,}u)\text{,}\dots \text{,}{G}_k(x\text{,}u))={\int }_a^{b}g(t\text{,}x\text{,}u)\mathit{dt}$ and $H:X\times Y\to {\mathbb{R}}^n$ by $H(x\text{,}u)=(H_1(x\text{,}u)\text{,}\dots \text{,}{H}_n(x\text{,}u))={\int }_a^b(h(t\text{,}x\text{,}u)-\dot{x})\mathit{dt}$. Mohan and Neogy [8] extended convexity to invexity for sets as follows: a set A is said to be invex with respect to $\eta :A\times A\to {\mathbb{R}}^n$ if $\forall x\text{,}y\in A$, $t\in [0\text{,}1]$, it is verified that $x+t\eta (y\text{,}x)\in A$. Observe that X and $K\subset X$ are invex.

The following convention for equalities and inequalities will be used. i. e., if $x=(x_1\text{,}\dots \text{,}{x}_n)\text{,}y=(y_1\text{,}\dots \text{,}{y}_n)\in {\mathbb{R}}^n$, then:

$$\begin{array}{cccc}x=y\hfill & \iff \hfill & x_i=y_i\text{,}\hfill & \forall i=1\text{,}\dots \text{,}n\text{,}\hfill \\ x<y\hfill & \iff \hfill & x_i<y_i\text{,}\hfill & \forall i=1\text{,}\dots \text{,}n\text{,}\hfill \\ x\leqq y\hfill & \iff \hfill & x_i⩽y_i\text{,}\hfill & \forall i=1\text{,}\dots \text{,}n\text{,}\hfill \\ x⩽y\hfill & \iff \hfill & x\leqq y\text{,}\hfill & \text{and there exists}i\text{such that}{x}_i<y_i.\hfill \end{array}$$

We will move forward in the study and localization of efficient solutions of (MCP), whose concept was introduced in mathematical programming problems by Pareto [9] and extended to multiobjective control problems, as showed in the following definition.

Definition 1

$(\overline{x}\text{,}\overline{u})\in K$ is said to be an efficient solution of (MCP) if there does not exist another $(x\text{,}u)\in K$ such that $F(x\text{,}u)⩽F(\overline{x}\text{,}\overline{u})$.

Chankong and Haimes [10] proved that $(\overline{x}\text{,}\overline{u})$ is an efficient solution for (MCP) if and only if $(\overline{x}\text{,}\overline{u})$ solves the k-related scalar control problems $({\mathrm{MCP}}_k(\overline{x}\text{,}\overline{u}))$ for all $k=1\text{,}2\text{,}\dots \text{,}p$, defined as

$$\begin{array}{cc}\hfill & \begin{array}{cc}({\text{MCP}}_k(\overline{x}\text{,}\overline{u}))\hfill & \text{Minimize}{\int }_a^b{f}_k(t\text{,}x\text{,}u)\mathit{dt}\hfill \\ \hfill & \text{subject to}:\hfill \\ \hfill & x(a)=\alpha \text{,}x(b)=\beta \text{,}\hfill \end{array}\hfill \\ \hfill & {\int }_a^b{f}_i(t\text{,}x\text{,}u)\mathit{dt}⩽{\int }_a^b{f}_k(t\text{,}\overline{x}\text{,}\overline{u})\mathit{dt}\text{,}i=1\text{,}\dots \text{,}p\text{,}i\ne k\text{,}\hfill \\ \hfill & g(t\text{,}x\text{,}u)\leqq 0\text{,}t\in I\text{,}\hfill \\ \hfill & h(t\text{,}x\text{,}u)=\dot{x}\text{,}t\in I\text{,}\hfill \end{array}$$

Under constraint qualification conditions, Chandra et al. [4] gave the Fritz–John necessary optimality conditions for the existence of an optimal solution for a scalar control problem (i.e., $p=1$ in (MCP)). Inspired in Mond and Hanson [11], Bhatia and Kumar [12], and Mishra and Mukherjee [13], and following Zhian and Qingkai [14], it follows the next vectorial Kuhn–Tucker optimality conditions.

Theorem 1 vectorial Kuhn–Tucker conditions

Let $(\overline{x}\text{,}\overline{u})\in K$ an efficient solution of (MCP), assume that constraint qualification conditions are satisfied [4] and $(\overline{x}\text{,}\overline{u})$ is an optimal solution of $({\mathit{MCP}}_k(\overline{x}\text{,}\overline{u}))$ and also normal for at least one $k\in \{1\text{,}2\text{,}\dots \text{,}p\}$. Then, there exist $\tau \in {\mathbb{R}}^p$ and piecewise smooth functions $\lambda :I\to {\mathbb{R}}^k$ and $\mu :I\to {\mathbb{R}}^n$ such that the following vector optimality conditions are fulfilled:

$${\tau }^T{f}_x(t\text{,}\overline{x}\text{,}\overline{u})+\lambda (t{)}^T{g}_x(t\text{,}\overline{x}\text{,}\overline{u})+\mu (t{)}^T{h}_x(t\text{,}\overline{x}\text{,}\overline{u})+\dot{\mu }(t)=0\text{,}$$

$${\tau }^T{f}_u(t\text{,}\overline{x}\text{,}\overline{u})+\lambda (t{)}^T{g}_u(t\text{,}\overline{x}\text{,}\overline{u})+\mu (t{)}^T{h}_u(t\text{,}\overline{x}\text{,}\overline{u})=0\text{,}$$

$$\lambda (t{)}^{T}g(t\text{,}\overline{x}\text{,}\overline{u})=0\text{,}$$

$$\tau \ne 0\text{,}(\tau \text{,}\lambda (t))⩾0$$

for all $t\in I$, except at discontinuities.

Proof

The procedure is similar to that for Theorem 6 in [14]. □

Mond et al. [15] extended the concept of invexity, given by Hanson [16], to continuous functions. Mond and Smart [5] defined the invexity for functionals instead of functions, and proved that invexity of F is necessary and sufficient for its critical points to be global minima, which coincides with the original concept of an invex function [16], [17] for mathematical programming problems. Similarly, in [18], [19], pseudoinvexity was extended to continuous functions and in [12] to functionals. In fact, the class of invex functionals and the class of pseudoinvex functionals are equivalent, such as it can be seen in [3]. Recently, these classes of functionals have been generalized to the vectorial case with the utilization of generalized invexity for vector functionals (see [20], [21], [22], [23], [24], [25]) and applied to symmetric duality or study of sufficient conditions for a Kuhn–Tucker point to be efficient solution in control and variational problems.

For scalar control problems ($p=1$), Mond and Smart [5] proved that the reciprocal of Theorem 1 holds if $(\int f\text{,}\int {\lambda }^{T}g\text{,}\int {\mu }^T(h-\dot{x}))$ is invex for any $\lambda (t)⩾0$, $\mu (t)$. Recently, Arana et al. [3] have showed by an example that this invexity condition is not necessary in order that all Kuhn–Tucker points are optimal solutions of a scalar control problem. They have moved forward the result given by Mond and Smart [5] by the introduction of the condition of KT-invex control problem, and have proved that KT-invexity for a control problem is characterized for all Kuhn–Tucker points to be optimal solutions. In the following section, we define a new condition on functionals of the multiobjective control problem (MCP): V-KT-pseudoinvexity, which is a natural extension of KT-invexity introduced by Arana et al. [3]. We establish that V-KT-pseudoinvexity is necessary and sufficient for all Kuhn–Tucker points to be efficient solutions, which generalizes characterization results given by Arana et al. [3]. In Section 3, we illustrate the nature of these results with an example.