Local Pareto optimality conditions for vector quadratic fractional optimization problems
There are several concepts and definitions that characterize and give optimality conditions for solutions of a vector optimization problem. One of the most important is the first-order necessary optimality condition that generalizes the Karush-Kuhn-Tucker condition. This condition ensures the existence of an arbitrary neighborhood that contains an local optimal solution. The present work we introduce an alternative concept to identify the local optimal solution neighborhood in vector optimization problems. The main aspect of this contribution is the development of necessary and sufficient Pareto optimality conditions for the solutions of a particular vector optimization problem, where each objective function consists of a ratio quadratic functions and the feasible set is defined by linear inequalities. We show how to calculate the largest radius of the spherical region centered on a local Pareto solution in which this solution is optimal. In this process we may conclude that the solution is also globally optimal. These conditions might be useful to determine termination criteria in the development of algorithms, including more general problems in which quadratic approximations are used locally.
keywords:Pareto optimality conditions ; efficient solutions ; vector optimization ; vector quadratic fractional optimization problems
In general the vector optimization problem (VOP) appears in the processes for decision-making and is represented as the following problem:
where , , are functions of real variables and is a nonempty subset. The fields of real functions contains . We denote by the feasible set that is the intersection of with the set of points in which , . If , we say that is a feasible point. is the result of the objective function if the decision maker chooses the action .
Let denote the nonnegative real numbers and denote the transpose of the vector . Furthermore, we will adopt the following conventions for inequalities among vectors. If and , then
if and only if , for all ;
if and only if , for all ;
if and only if , for all ;
if and only if and .
Similarly we consider the equivalent convention for inequalities and .
We say that dominates in (VOP) if , that is a neighborhood of , that is an open ball centered on , the radius and the boundary , and that is a closed ball, defined by Euclidean distance.
Different optimality notions for the problem (VOP) are referred to as a Pareto optimal solution Livro:Pareto, two of which are defined as follows.
A feasible point is said to be a (locally) Pareto-efficient optimal solution of (VOP), if there does not exist another () such that .
A feasible point is said to be a (locally) weakly Pareto-efficient optimal solution of (VOP), if there does not exist
another () such that .
Every Pareto-efficient optimal solution is a weakly Pareto-efficient optimal solution. The Pareto-efficient optimal solution set is denoted by Eff, and locally Pareto-efficient optimal solution by Leff. The Pareto-efficient optimal solutions are also known as globally Pareto-efficient optimal solutions. Let the set Leff, we say that Leff is the Pareto-optimal curve.
There are many contributions, concepts, and definitions that characterize and give the Pareto-efficient optimality conditions for solutions
of a vector optimization problem (see, for instances Livro:Chankong; Livro:Jahn; Livro:Miettinen; Tese:Osuna; Livro:Romero). One of the most
important is the first-order necessary optimality condition that generalizes the Karush-Kuhn-Tucker (KKT) condition. However, to obtain the
sufficient optimality conditions, it is necessary to impose additional assumptions (like convexity and its generalizations) in the objective
functions and in the constraint set. Otherwise, is only possible to characterize the locally Pareto-efficient optimal solutions. In general
the non-equivalence between locally and globally Pareto-efficient optimal solutions is a difficult problem to be faced. The following example
shows a simple problem which illustrates this. We recall that denote the gradient of the function at point .
Exemplo: Consider the following problem:
which their objective functions can be expressed as
where , , , , and whose matrices and are
This problem is an unconstrained two objectives optimization problem, which the objectives are quadratic functions with indefinite Hessian matrices. If , it can not simultaneously occurs
In fact, let . If and , then
This is a contradiction, therefore inequalities (1) only occur if . A necessary condition for a point be a locally Pareto-efficient optimal solution Livro:Chankong is that there are real numbers , not all zero, such that
If and , then . In the direction , for every point of the form , for each real value , simultaneously occur and . Therefore, the point is not a locally Pareto-efficient optimal solution.
If and , then . In the direction , for every point of the form , for each real value , simultaneously occurs and . Therefore, the point is not a locally Pareto-efficient optimal solution.
Now, if , an equivalent way to write the condition (2) is
But, this condition is also sufficient for the points , unlike and , are locally Pareto-efficient optimal solution. In fact, suppose that is not locally Pareto-efficient optimal solution, then there exists a direction and a number real such that is valid. That is, there is a descent direction which locally occurs and , with and . However, for and if the equation (3) is valid, by Stiemke’s alternative theorem Art:stiemke1915positive, and has no solution, and and has no solution.
Thus, when and Stiemke’s alternative theorem are valid to must occur and , and to must simultaneously occurs and , or and . What is impossible, because (1). Therefore, the locally Pareto-efficient optimal solutions are those that satisfy (3), that is, and , that is,
The points satisfying (4) are represented in Figure 1. The locally Pareto-efficient optimal solutions to this problem are grouped into three disconnected branches: , and . In addition, it is possible to verify that only the solutions set generated by inequality are globally Pareto-efficient optimal solutions (branch plotted in the graph). We accept this by looking Figure 2. It illustrates the graph in the plane representing the Pareto-optimal curve, that is, the branches images , and , by functions and plotted in the coordinates . , and represent the branches images , and , respectively. We see in Figure 2 that the locally Pareto-efficient optimal solutions set belonging to the branch also are globally Pareto-efficient optimal solutions, that is, globally non-dominated.
We have just seen a simple example which the globally Pareto-efficient optimal solutions set is strictly contained in the locally Pareto-efficient optimal solutions set, but investigate the locally or globally efficient optimal solutions are not a trivial task. In this paper, we present new ways that can facilitate this analysis.
This paper is organized as follows. We start by defining some notations and basic properties. In Section 3, the new concept of radius of efficiency and the relationships among associated problem are presented. In Section 4, using this concept some necessarily and sufficient Pareto optimality conditions are established. Finally, comments and concluding remarks are presented In Section 5.
2 Quadratic fractional problem and literature review
In this paper, we deal with a particular case of (VOP), which each objective function consists of a ratio of two quadratic functions. Without generalized convexity assumptions in the objective functions, we show how to calculate the largest radius of the spherical region centered on a local Pareto-efficient solution in which this solution is optimal. Let us consider the following vector quadratic fractional optimization problem:
where and , , are quadratic functions of real variables. In addition, we assume that , , for any in nonempty . The fields of real functions contains . We denote by the feasible set that is the intersection of with the set of points in which , . We choose the functions that preserves the signal. We denote the unconstrained (VQFP) by (VQFP).
The fractional optimization problems arise frequently in the decision making applications, including science management, portfolio selection, cutting and stock, game theory, in the optimization of the ratio performance/cost, or profit/ investment, or cost/time and so on.
There are many contributions dealing with the scalar (single-objective) fractional optimization problem (FP) and vector fractional optimization problem (VFP). In most of them, using convexity or their generalizations, optimality conditions in the KKT sense and the main duality theorems for optimal points are obtained. With a parametric approach, which transforms the original problem in a simpler associated problem, Dinkelbach Art:Dinkelbach, Jagannathan Art:Jagannathan and Antczak Art:Antczak establish optimality conditions, presents algorithms and apply their approaches in a example (FP) consisting of quadratic functions. For example, Dinkelbach Art:Dinkelbach presents some theoretical results that relate the two problems, and proposes an algorithm that converges to the minimum of problem (FP) to perform a sequence of operations in associated parametric problem. Using some known generalized convexity, Antczak Art:Antczak, Khan and Hanson Art:KhanAndHanson, Reddy and Mukherjee Art:ReddyAndMukherjee, Jeyakumar Art:Jeyakumar, Liang et al. Art:LiangAndHuangAndPardalos establish optimality conditions and theorems that relate the pair primal-dual of problem (FP). In Craven Art:Craven1981; Livro:Craven1988 and Weir Art:Weir, other results for the scalar optimization (FP) can be found.
Further, Liang et al. Art:LiangAndHuangAndPardalosMultiobjetivo extended their approach to the vector optimization case (VFP) considering the type duals of Mond-Weir Art:MondWeirDual, Schaible Art:SchaibleDual1; Art:SchaibleDual2 and Bector Art:BectorDual. Considering the parametric approach of Dinkelbach Art:Dinkelbach, Jagannathan Art:Jagannathan, Bector et al. Art:BectorAndChandraAndSingh and two classes of generalized convexity, Osuna-Gómez et al. Art:Osuna establish weak Pareto-efficient optimality conditions and the main duality theorems for the differenciable vector optimization case (VFP). Santos et al. Art:Lucelina deepened these results to the more general non-differenciable case (VFP). Jeyakumar and Mond Art:JeyakumarAndMond use generalized convexity to study the problem (VFP) and with the parametric approach, Singh and Hanson Art:SinghAndHanson extended the results obtained by Geoffrion Art:Geoffrion.
Few studies are found involving quadratic functions at both the numerator and denominator in the ratio objective function. Most of them involve the mixing of linear and quadratic functions. The closest approaches of the scalar quadratic fracional optimization problem (QFP) are considered by Crouzeix et al. CrouzeixAndFerlandAndSchaible, Schaible and Shi Art:SchaibleAndShiQuadratic, Gotoh and Konno Art:GotohAndKonno, Lo and MacKinlay Art:LoAndMacKinlay and Cambini et al. CambiniAndCrouzeixAndMartein. On the other hand, Benson Art:BensonQuadratic considered a pure (QFP) consisting of the convex function, develop some theoretical properties and optimality conditions, he presents an algorithm and its convergence properties.
The closest approaches of the vector optimization case (VQFP) are considered by Beato et al. Art:BeatoAndRuizAndLuqueAndBlanquero; Art:BeatoAndInfanteAndRuiz, Arévalo and Zapata Art:ArevaloAndZapata, Konno and Inori KonnoAndInori, Rhode and Weber Livro:RhodeAndWeber, Kornbluth and Steuer Art:KornbluthAndSteuer, Korhoen and Yu Art:KorhonenAndYu; Art:KorhonenAndYu2. Using an iterative computational test, Beato et al. Art:BeatoAndRuizAndLuqueAndBlanquero; Art:BeatoAndInfanteAndRuiz characterize the Pareto-efficient optimal point for the problem (VQFP), consisting of a linear and quadratic functions, and using the function linearization approach of Bector et al. Art:BectorAndChandraAndSingh, some theoretical results are obtained. Arévalo and Zapata Art:ArevaloAndZapata, Konno and Inori KonnoAndInori, Rhode and Weber Livro:RhodeAndWeber analyze the portfolio selection problem. Kornbluth and Steuer Art:KornbluthAndSteuer use an adapted Simplex method in the problem (VFP) consisting of linear functions. Korhonen and Yu Art:KorhonenAndYu; Art:KorhonenAndYu2 propose an iterative computational method for solving the problem (VQFP), consisting of the linear and quadratic functions, based on search directions and weighted sums.
The approach taken in this work is different from the previous ones. We believe that the approach presented here facilitates the resolution of the problem (VQFP). The main aspect of this contribution is the development of necessary and sufficient Pareto-efficient optimality conditions for a particular vector optimization problems based on the calculation of largest radius of the spherical region centered on a local Pareto-efficient solution in which this solution is optimal. In this process we may conclude that the solution is also globally Pareto-efficient optimal. These conditions might be useful to determine termination criteria in the development of algorithms, including more general problems in which quadratic approximations are used locally.
3 Radius of efficiency
We introduce the new concept radius of efficiency for (VOP) and present some Pareto-efficient optimality conditions for (VQFP) from this alternative approach, and then we extend the results to the constrained case (VQFP).
3.1 Radius of efficiency in the (VOP)
From certain particular properties verified in some search directions of the objective functions, we want to detect when a local Pareto-efficient optimal solution is equivalent to a global Pareto-efficient optimal solution. With this approach is possible to identify a spherical region of feasible points where a local Pareto-efficient optimal solution is not dominated.
We say that a point is -efficient or has radius of efficiency in (VOP) if Leff and there does not exist another point that dominates .
Note an important difference between the locally Pareto-efficient optimal solution definition and the radius of efficiency definition of a locally Pareto-efficient optimal solution. In the first case, from a theoretical point of view we know that always there is an arbitrary neighbourhood of , where is not dominated. Naturally, this neighbourhood always can be regarded as a ball of arbitrary radius in . In the second case, is possible to calculate the largest radius of the spherical region such that is not dominated in , or is possible to conclude that is not dominated everywhere the feasible set .
Some important reasons for the use of the radius of efficiency in the problems (VOP) can be indicate. We always can consider a local solution in a fixed well-know spherical region instead in an arbitrary spherical region. We can determine a subset compact where there does not exist another points that dominate a specific solution, what is useful in the resolution of (VOP), because if the decision maker knows the radius of efficiency, he can estimates the cost to try to find a new solution, and also choose a more suitable search procedure. Auxiliary problems induced by the concept of radius of efficiency can be used to conclude the global Pareto-efficiency.
Naturally, if is -efficient, then it is -efficient, . Similarly we say that is -efficient if it is efficient in .
3.2 Radius of efficiency in the (VQFP)
We calculate the radius of efficiency of a solution and some results from the calculation are presented. By hypothesis, we assume that Leff. To determine the radius of efficiency for this solution , we must ask: What is the smallest value of such that among every unitary direction in which is valid? The answer to this question provides the maximum radius of efficiency of .
The next theorems allow us to characterise when a locally Pareto-efficient optimal solution is equivalent to a Pareto-efficient optimal solution. We will use these theorems to identify the maximum radius of efficiency and analyse the dominance of a locally Pareto-efficient optimal solution in the feasible set.
Similarly to Dinkelbach Art:Dinkelbach and Jagannathan Art:Jagannathan, which transform the fractional optimization in a new problem, we consider the following problem associated with the (VQFP).
were and , , , , , are the same functions defined in (VQFP).
The (VQFP) is similarly introduced by Osuna-Gómez et al., which derive optimality conditions and duality results for the weakly Pareto-efficient optimal solutions. The following theorem and its proof is an approach equivalent to Lemma 1.1 presented in Art:Osuna, but we consider the Pareto-efficient optimal solutions.
Leff if and only if Leff. In addition, is locally Pareto-efficient optimal solution for in if and only if is locally Pareto-efficient optimal solution for in .
Proof Let and be locally Pareto-efficient optimal solution for in . Suppose that Leff, then there exists another point satisfying
Which contradicts Leff in , and therefore Leff in .
Similarly, let and be locally Pareto-efficient optimal solution for in . Suppose that Leff, then there exists another point , satisfying
Which contradicts Leff in , and therefore Leff in .
Note that on associated problems, for each Leff we can only ensure that Leff Leff.
The same arbitrary neighborhood equivalent to a ball of arbitrary radius in centered at appears in both problems (VQFP) and (VQFP), and we want to compute a neighborhood of maximum radius , such that Leff in . Then, calculate the radius of efficiency of the solution in the (VQFP) is equivalent to calculate the radius of efficiency of the solution in the (VQFP), and so we can choose between the two problems one whose calculation is easier.
4 Optimality Conditions from Radius of efficiency
First the optimality conditions are established for the unconstrained problem and then we extend the results for the feasible set defined by linear inequalities.
4.1 Radius of efficiency in the unconstrained case
For each and all we consider the objective functions defined as and where , , symmetric, symmetric and positive semidefinite, , and , , with , where is the solution of the system , that is, is the point where the function reaches its minimum and this ensures that , . We cannot consider cases where has no solution. Similarly, we denote by (VQFP) the unconstrained (VQFP). We recall that denote the Hessian matrix of the function at point .
Further, we define some sets and parameters that are used throughout this work. Given , we define the following quadratic function
and given an unitary direction , we define
The functions , , defined in (5) are the same objective functions of the (VQFP) unless the constants term , and so . We denote by the number of elements in the set .
For a better understanding of sets and parameters above, consider the Taylor expansion around zero of the function , , . Since is a quadratic function, we obtain
In other words, is a real function of one variable, whose graph is a parable, and has constant term , linear term and quadratic term . Thus, it becomes easier to interpret the sets , and . Given an unitary direction , the set is formed by the indices whose functions decrease to values close and are convex, while the set is formed by the indices whose functions grow to values close and are concave.
Figure 3 shows typical elements of the sets and , in which the graphs represent two functions , and are plotted the points of function in the coordinates for , which are two parables: a convex indicating that the function index belongs to the set (where, and ); and a concave indicating that the function index belongs to the set (where, and ).
The set is formed by the indices where each quadratic function has non-negative linear term and positive quadratic term or has positive linear term and non-negative quadratic term.
Figure 4 shows typical elements of the set and , in which the graphs represent three functions . In relation the function index belongs to the set , two examples are plotted: a convex parable (where, and ); and a growing straight line (where, and ).
The parameter is defined as the maximum of the positive roots of the equations in which indices belong to . Similarly, if , the parameter is defined as the minimum of the positive roots of the equations in which indices belong to . If , we define .
For each index , another root of the equation is , but we are interested in the values . Figure 3 shows these parameters, the parameter and are the crossover point between the curve and the dotted straight line for and , respectively. Figure 4 shows a positive root of the equation for .
Given an unitary direction , the sets , and are real intervals contained in , where is the extreme left value of the interval and is the extreme right value of the interval . When occur , they are the end points of the interval and if occur , we obtain . However, in some particular cases, for example when occur and , we obtain , that is , but the most suitable in this case is set . This case and others like it are important and are explained in detail in due course.
Figure 3 illustrates examples of the intervals , and . In it, we represent over the dotted straight line the interval , over the -axis the interval , and below the -axis the interval . When we choose , because when we always obtain . Note that in Figure 3 we have for all and . On the other hand, Figure 6 illustrates an example in which .
Before the next result, we present some important details in order to understand the possibility of obtaining an unitary direction and a constant , in which is valid for a solution Leff, and what are their relations with the sets , and .
Consider the Taylor expansion around zero of the each function and , , in solution and along the direction ,
Performing some manipulations, we obtain