Multiobjective optimization is nowadays a word of order in engineering projects. Although the idea involved is simple, the implementation of any procedure to solve a general problem is not an easy task. Evolutionary algorithms are widespread as a satisfactory technique to find a candidate set for the solution. Usually they supply a discrete picture of the Pareto front even if this front is continuous. In this paper we propose three methods for solving unconstrained multiobjective optimization problems involving quadratic functions. In the first, for biobjective optimization defined in the bidimensional space, a continuous Pareto set is found analytically. In the second, applicable to multiobjective optimization, a condition test is proposed to check if a point in the decision space is Pareto optimum or not and, in the third, with functions defined in
Life is about making decisions and the choice of the optimal solutions is not an exclusive subject for scientists, engineers, and economists. Decision making is present in daytoday life. Looking for an enjoyable vacancy, everyone will formulate an optimization problem to a travel agent, a problem like with a minimum amount of money visit a maximum number of places in a minimum amount of time and with the maximum level of comfort. Usually all real design problems have more than one objective; namely, they are multiobjective. Moreover, the design objectives are often antagonistic.
Edgeworth [
Few years later, in 1896, Pareto [
Since then, many researchers have been dedicated to developing methods to solve this kind of problem. Interestingly, solutions for problems with multiple objectives, also called multicriteria optimization or vector optimization, are treated as Pareto optimal solutions or Pareto front, although, as Stadler [
Extensive reviews of multiobjective optimization concepts and methods are given by Miettinen [
Thanks to the computer development, optimization of large scale problems became a common task in engineering designs. The development of high speed computers and their increasing use in several industrial branches led to significant changes in the design processes. Currently, the computers, each time faster, allow the engineer to consider a wider range of design possibilities and optimization processes allow systematic choice between alternatives, since they are based on some rational criteria. If used adequately, these procedures can, in most cases, improve or even generate the final results of a design.
Associated with computer development, many of the research done in optimization is focused on numerical methods to solve any kind of problem, but sometimes simplified problems can give important clues to the designer during the tradeoff phases of a decision.
The present work aims to bring new approaches to solve multiobjective optimization problems, providing a rapid solution for the Pareto set if the objective functions involved are quadratic.
The rest of the paper is organized into 3 sections. In the first section a general multiobjective optimization problem is formulated and the nature of optimal solutions from the Pareto perspective and the necessary conditions to be met are defined. In the second section, three propositions are done to solve the unconstrained multiobjective optimization problems involving quadratic functions. In the first section the general problem comprises two bidimensional functions. In this case, the proposition permits to find the Pareto front analyticaly. In the second section, the problem considers the minimization problem with three or more functions, keeping the decision space in two dimensions. In this case the proposition helps to find the Pareto points and their boundary in the decision space. In the third section, proposition of the decision space is expanded to any dimensional size. Finally, a section with the conclusions and the proposed future work is presented.
Multiobjective optimization problems (MOOP) can be defined by the following equations:
The notion of “optimum” in solving problems of multiobjective optimization is known as “Pareto optimal.” A solution is said to be Pareto optimal if there is no way to improve one objective without worsening at least another; that is, the feasible point
In fact, optimizing multiobjective problems expressed by (
Equations (
The methods we will propose in the next sections can be classified in posteriori preference articulation and an extensive literature review of the most important methods to solve multiobjective optimization problems can be found in Augusto et al. [
In this section we propose a simple strategy to determine the Pareto set in the decision space and the corresponding Pareto front in the function space, for MOOP involving two bidimensional differentiable functions. Consider an unconstrained multiobjective optimization problem. From (
If there exists a Pareto front for the minimization problem with two continuous and differentiable functions defined in
As the gradients of each function are orthogonal to contours and point outwards from the minimum, the curve mentioned in Proposition
Graphical representation of Proposition
Proposition
minimize:
Applying the optimality condition,
As the system of (
Function gradients
Therefore, (
As an example, let us consider the following biobjective problem:
minimize:
In Figure
Graphical representation of Proposition
Continuous Pareto set obtained by the proposed method
Continuous Pareto front, the Pareto set image in the function space
Pareto set for the performance functions
Pareto front for performance functions
For comparison, it is shown in Figures
In the previous section we found a closed form solution for the optimization of two quadratic functions in the bidimensional decision space. Unfortunately, we did not find a similar solution when we add more functions in the problem. Nevertheless, the idea behind Proposition
Consider an optimization problem involving three continuous differentiable functions
In Figure
Pareto optimality condition for three or more functions in
An equilibrium condition exists when
Based on this idea we suggest the following.
Let
The direction of
Equations (
In Figure
Pareto optimality condition applied to the threeobjective optimization problem involving functions defined in the twodimensional decision space.
Pareto set
Pareto front
Pareto front
Pareto front
Pareto front
In the former two sections we have considered unconstrained MOOP with quadratic functions defined in the twodimensional space. To proceed to larger dimensions, let us define a quadratic function in the
Using (
With these definitions, let
As
In (
Let us assume that all
Consequently, the Pareto optimum solution
In this approach, we have considered that
Consider a MOOP involving
Sort, at random over the interval
Perform a normalization such as
Calculate
Solve the linear system
Repeat steps (1) to (4) for the number
Even requiring solutions of
Before advancing to the applications, consider an ellipsoid enclosed in a parallelepiped of sizes
Representation of an ellipsoid, a quadratic function
The family of quadratic functions that represents this ellipsoid can be written as follows:
The ellipsoid can be rotated around the
The local coordinate system can be positioned at a point
To get the transformation matrix
With the previous definitions, consider the following unconstrained MOOP:
Coefficients for objective functions
Function  Semiaxis  Rotation  Origin  









 

1  2  3  0  0  − 
10  10  0 

1  2  3  0  0  0  0  −10  0 

1  2  3  0  0 

−10  10  0 
Solution of the unconstrained MOOP with the quadratic functions defined in Table
Three quadratic functions,
Pareto set
Pareto set,
Pareto front
The Pareto set for this problem, illustrated in Figure
As all ellipsoids were placed over
The Pareto front is shown in Figure
In the next example three ellipsoids with different orientations, as defined in Table
Optimization problem with 3 objective functions.
Function  Semiaxis  Rotation  Origin  









 

1  2  3  0  0  0  0  0  0 

1  2  3  0 

0  10  0  0 

1  2  3  0  0 

0  10  10 
The Pareto set of this optimization problem found by the proposed methodology delineates the curved surface shown in Figure
Solution of the unconstrained MOOP with the quadratic functions defined in Table
Pareto set
Pareto front
Adding to the unconstrained MOOP the function
Optimization problem with 4 objective functions.
Function  Semiaxis  Rotation  Origin  









 

1  2  3  0  0 

0  0  0 

1  2  3  0 

0  15  0  0 

1  2  3  0  0 

0  15  0 

1  2  3  0  0  0  10  10  15 
Pareto set of the unconstrained MOOP with the quadratic functions defined in Table
(a) Rotation
In the problems all functions were defined by convenience in
Most of the real problems are multiobjective with their objective functions being antagonistic. To solve this problem many researchers are developing methods to solve multiobjective optimization problems without reducing them to single objective. Up to now, evolutionary algorithms are widespread as a general technique to find a candidate set of the optimal solutions. These algorithms provide a discrete picture of the Pareto front in the function space, without bringing too much information about the decision space.
In the framework of this paper, we have proposed different methods to determine the Pareto set of unconstrained multiobjective optimization problems involving quadratic objective functions. Three different procedures were proposed. One for biobjective optimization, with functions defined in
It is apparent that the Pareto set for two distinct twodimensional functions is a curve, and for three and above, the Pareto set is a surface. In threedimensional space, for two distinct threedimensional functions, the Pareto set will be a space curve; for three functions, a surface; and for four functions and above, a solid. Although the proposed methods are restricted to unconstrained optimization, the authors believe they can be extended to constrained problems and are working on it.
See Figures
Decision maker
Objective functions vector
Genetic algorithm
Number of objective functions
KarushKuhnTucker
Number of equality constraint functions
Number of inequality constraint functions
Multiobjective optimization problem
Nondominated sorting genetic algorithm, version two
Dimension of the design space
Function or criterion space
Decision variables or design space
Feasible region in the design space
Decision variable vector
Nondominated solution of a multiobjective optimization problem
Lower and upper bounds of the design space
Weighting factor for the
Vector of
Weighting factor for
Vector of
Weighting factor for
Vector of
Gradient operator.
The authors declare that there is no conflict of interests regarding the publication of this paper.