A decomposition approach to nonlinear multi-criteria optimization/equilibrium problems

Transcription

1 A decomposition approach to nonlinear multi-criteria optimization/equilibrium problems Nicolae Popovici popovici Babeş-Bolyai University, Cluj-Napoca, Romania RISM4, September 14-18, 2015, Varese, Italy 1 / 56

2 Introduction A natural way of approaching a complex problem is to decompose it into simpler problems of the same or similar nature. By combining the celebrated Theorem 1 (Carathéodory, 1911, Rend. Circ. Mat. Palermo) If S R n is nonempty, then every point x 0 convs can be expressed as a convex combination of at most n + 1 elements of S. with the following (not so well known) result Theorem 2 (Fejér, 1922, Math. Ann.) Consider a finite set S R 2 (Euclidean plane). For any x 0 R 2 we have: x 0 convs iff x R 2 s.t. x s < x 0 s s S. the idea of decomposing multi-criteria optimization problem arises. 2 / 56

3 Outline 1 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility 2 Equilibrium conditions via scalarization 3 / 56

4 Multi-criteria optimization problems Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Let S be a nonempty set. For any functions f 1,..., f m : S R (m N, m 2) we can formulate various (actually 2 m ) multi-criteria optimization problems of type: f 1 (x) min (max)... (P) f m (x) min (max) s.t. x S. 4 / 56

5 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility As in scalar optimization, by changing the sign of certain functions if necessary, we can always reduce our study to a multi-criteria minimization problem Minimize f (x) = (f 1 (x),..., f m (x)) s.t. x S or, alternatively, to a multicriteria maximization problem Maximize f (x) = (f 1 (x),..., f m (x)) s.t. x S. 5 / 56

6 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Example 3 (Location Problem) Let d : R n R n R be a distance function, as for instance d(x, y) = x y for a certain norm : R n R. Let A := {a 1,..., a m } R n be a set of points with carda = m 2 (representing a priori given facilities when n = 2). Then f 1 (x) := d(x, a 1 ) min... f m (x) := d(x, a m ) min x S = R n is called multi-criteria single-facility location problem. 6 / 56

7 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility In this example the functions f 1,..., f m have no common minimizer, i.e., the scalar optimization problems f i (x) := d(x, a i ) min i {1,..., m}, x S = R n, have no common optimal solutions, since m i=1 argmin f i (x) = x S m {a i } =. i=1 So, multi-criteria optimization requires appropriate concepts of optimality. The first reference to address conflicting objectives is attributed to Pareto (1906, Manuale di Economia Politica, Milano). 7 / 56

8 Optimality concepts Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Consider three binary relations on R m, defined for any u = (u 1,..., u m ), v = (v 1,..., v m ) R m as follows: u v : u i v i, i {1,..., m}; u < v : u i < v i, i {1,..., m}; u v : u v and u v i {1,..., m}, u i v i ; j {1,..., m} s.t. u j < v j. These notations are consistent with the standard ones when m = 1. Of course, in this case, < and coincide. 8 / 56

9 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Given a nonempty subset Y of R m, we introduce the sets of (Pareto) minimal, weakly minimal, (Pareto) maximal, and weakly maximal points of Y, respectively by Min Y := {y 0 Y y Y s.t. y y 0 }; WMin Y := {y 0 Y y Y s.t. y < y 0 }; Max Y := {y 0 Y y Y s.t. y y 0 }; WMax Y := {y 0 Y y Y s.t. y > y 0 }. It is easily seen that Min Y WMin Y bd Y ; Max Y WMax Y bd Y. 9 / 56

10 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Definition 4 The set of efficient solutions (Pareto optimal solutions) of the multi-criteria minimization problem is defined as Min(S f ) := f 1 (Min f (S)) = {x 0 S x S : f (x) f (x 0 )}. The set of weakly efficient solutions (Slater optimal solutions) of the multi-criteria minimization problem is given by WMin(S f ) := f 1 (WMin f (S)) = {x 0 S x S : f (x) < f (x 0 )}. 10 / 56

11 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Definition 5 The set of efficient solutions (Pareto optimal solutions) of the multi-criteria maximization problem is defined as Max(S f ) := f 1 (Max f (S)) = {x 0 S x S : f (x) f (x 0 )}. The set of weakly efficient solutions (Slater optimal solutions) of the multi-criteria maximization problem is given by WMax(S f ) := f 1 (WMax f (S)) = {x 0 S x S : f (x) > f (x 0 )}. 11 / 56

12 Optimality conditions Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Theorem 6 (sufficient conditions for efficiency) If c = (c 1,..., c m ) R m and c > 0 m, then argmin(c 1 f 1 (x) + + c m f m (x)) Min(S f ); x S argmax(c 1 f 1 (x) + + c m f m (x)) Max(S f ). x S Theorem 7 (sufficient conditions for weak efficiency) If c = (c 1,..., c m ) R m and c 0 m, then argmin(c 1 f 1 (x) + + c m f m (x)) WMin(S f ); x S argmax(c 1 f 1 (x) + + c m f m (x)) WMax(S f ). x S 12 / 56

13 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Theorem 8 (necessary & sufficient conditions for weak efficiency) Assume that S is a convex set. 1 If all functions f 1,..., f m are convex, then WMin(S f ) = argmin c R m,c 0 x S m 2 If all functions f 1,..., f m are concave, then WMax(S f ) = argmax c R m,c 0 x S m (c 1 f 1 (x) + + c 1 f m (x)). (c 1 f 1 (x) + + c 1 f m (x)). 13 / 56

14 Decomposition Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Let f = (f 1,..., f m ) : S R m be defined on a nonempty set S. Denote I m := {1,..., m}. For each I I m with I = k > 0, consider f I = (f i1,..., f ik ) : S R k, where I = {i 1 < < i k }. The optimization problem (P I ) Minimize s.t. x S. f I (x) can be seen as a subproblem of (P). Notice that (P I ) is a scalar optimization problem if I is a singleton. In this case the sets of efficient solutions and weakly efficient solutions of this subproblem, Min(S f I ) and WMin(S f I ), coincide with argmin x S f (x). 14 / 56

15 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility If I J I m, then we have WMin(S f I ) WMin(S f J ). In particular, WMin(S f I ) WMin(S f ). However, in general Min(S f I ) Min(S f J ). 15 / 56

16 Pareto reducibility Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Definition 9 (Popovici, 2005) The multi-criteria optimization problem (P) is said to be Pareto reducible if its weakly efficient solutions are efficient solution (i.e., Pareto optimal solutions) for the problem itself or for one of its subproblems, i.e., WMin(S f ) = Min(S f I ). =I I m Simple examples show that there are bicriteria optimization problems which are not Pareto reducible. An interesting research topic is to find sufficient/necessary conditions for Pareto reducibility. 16 / 56

17 Convex problems Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Theorem 10 (Lowe, Thisse, Ward & Wendell, 1984) Convex multi-criteria optimization problems are Pareto reducible. Corollary 11 Let be a norm on R n. For any nonempty set A := {a 1,..., a m } R n, the location problem Minimize f (x) = ( x a 1,..., x a m ) s.t. x R n is Pareto reducible. Can we say more for this special class of convex multi-criteria problems? 17 / 56

18 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility For every i {1, 2,..., m} and r > 0 let us denote B(a i, r) := {x R n x a i < r} = {x R n f i (x) < r}; B(a i, r) := {x R n x a i r} = {x R n f i (x) r}; S(a i, r) := {x R n x a i = r} = {x R n f i (x) = r}. For any x 0 R n the following hold: m m 1 x 0 Min(R n f ) B(a i, f i (x 0 )) = S(a i, f i (x 0 )). 2 x 0 WMin(R n f ) i=1 m i=1 i=1 B(a i, f i (x 0 )) =. 18 / 56

19 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Example 12 Let n = 2, m = 3, and x = (x 1 ) 2 + (x 2 ) 2 (Euclidean norm). Figure 1 : x 0 Min(R 2 f ) = WMin(R 2 f ) = conv{a 1, a 2, a 3 } 19 / 56

20 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Here is the explanation: Theorem 13 (Durier-Michelot, 1986) 1 If the norm is strictly convex, i.e., 1 2 (x + y) < 1 for all distinct points x, y R n with x = y = 1, then Min(R n f ) = WMin(R n f ). 2 If either the norm is generated by an inner product, i.e., x 2 =< x, x >, x R n, or n = 2 and the norm is strictly convex, then Min(R n f ) = Min(R n f ) = conv A = conv{a 1,..., a m }. 20 / 56

21 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Example 14 Let n = 2, m = 3, a 1 = (0, 0), a 2 = (1, 3), a 3 = (4, 3 2 ), and x = x 1 + x 2 (Minkowski norm). Figure 2 : x 0 Min(R 2 f ) = [0, 1] [0, 3 2 ] [1, 4] { 3 2 } {1} [ 3 2, 3]. 21 / 56

22 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Figure 3 : x 0 WMin(R 2 f ) = [0, 4] [0, 3]. 22 / 56

23 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Figure 4 : Computer generated solution sets, using an algorithm by Alzorba, Günther, Popovici & Tammer, / 56

24 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Non-convex problems We say that C R m is a cone if 0 R + C. A cone C is said to be pointed if C C = {0}. It is well known that a cone C is a convex set if and only if C + C = C. Definition 15 (Benoist & Popovici, 2001) Given a cone C in R m, a subset A of R m is called C-radiant if ray(a, a ) := a + R + (a a) A for all a, a A, a a + C. In particular, A is R m +-radiant if and only if for all a, a A with a a we have ray(a, a ) A. If A is upward, i.e., A = A + R m +, then A is R m +-radiant, but WMinA may be not R m +-radiant. 24 / 56

25 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Let X be a real linear space. For any points x, x X denote ]x, x [ := { (1 t)x + tx t ]0, 1[ }. Definition 16 Let S X be a convex set. A function ϕ : S R is called: quasiconvex if for all x, x S and x ]x, x [ we have ϕ(x) max{ϕ(x ), ϕ(x )}; semistrictly quasiconvex, if for all x, x S such that ϕ(x ) ϕ(x ) and for all x ]x, x [ we have ϕ(x) < max{ϕ(x ), ϕ(x )}. 25 / 56

26 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Theorem 17 (Popovici, 2005) Let f : S R m be defined on a nonempty set S X. If WMin (f (S) + R m +) is R m +-radiant, then the multi-criteria minimization problem (P) is Pareto reducible. Corollary 18 (Popovici, 2005, 2006) Let f = (f 1,..., f m ) : S R m be a function defined on a nonempty convex set S X. If f 1,..., f m are semistrictly quasiconvex and upper semicontinuous along line segments, then problem (P) is Pareto reducible. If f 1,..., f m are also lower semicontinuous along line segments, then WMin(S f ) = I Im I n+1 Min(S f I ). 26 / 56

27 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility The previous result is relevant for linear fractional functions of type f i (x) = a i, x + α i b i, x + β i, where a i, b i R n, α i, β i R, (b i, β i ) (0, 0), and S is a nonempty convex set such that S {x R n b i, x + β i > 0} for any i {1,..., m}. Such functions naturally occur in optimization problems involving criteria that are ratios, such as return on investment, dividend coverage and productivity measures (see, e.g., Cambini & Martein 2005, Choo & Atkins 1982, or Stancu-Minasian 1992). 27 / 56

28 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility Figure 5 : Three linear-fractional criteria 28 / 56

29 Optimality concepts (efficiency and weak efficiency) Optimality conditions via scalarization Decomposition and Pareto reducibility For more general functions, the structure of solution sets can be very sophisticated (see Benoist & Popovici 2000) Figure 6 : The homeomorphic image of Minf (S) by orthogonal projection onto the Bonnisseau-Cornet hyperplane 29 / 56

30 Scalar equilibrium problems Equilibrium conditions via scalarization Equilibrium problems have been intensively studied in the last two decades due to their wide range of applications. Actually, optimization problems, variational inequalities, saddle point (minimax) problems, Nash equilibria, complementarity problems, and other important problems, can be seen as particular instances of the general equilibrium problem (Blum & Oettli 1994; Iusem & Sosa 2001). The classical (scalar) equilibrium problem, as introduced by Muu & Oettli (1992), is governed by a real-valued bifunction f : A B R, where A and B are nonempty sets. It consists of finding an element x A, which satisfies f (x, y) 0, y B. 30 / 56

31 Equilibrium conditions via scalarization The vector equilibrium problems are governed by bifunctions which take values in a partially ordered real vector space. They have been investigated by many authors, beginning with Ansari (1996, 2000), Bianchi et al. (1997) and Oettli (1997). We consider the particular class of vector equilibrium problems which are governed by bifunctions whose outcome space is a finite-dimensional real Euclidean space, endowed with the usual componentwise ordering. This particular setting allows us to decompose any vector equilibrium problem into a family of equilibrium subproblems, each of them being governed by a bifunction obtained from the initial one by selecting some of its scalar components. 31 / 56

32 Equilibrium conditions via scalarization An appropriate framework for decomposition Then, similarly to multi-criteria optimization, we investigate the relationship between three types of solutions of these equilibrium subproblems, namely, weak, strong and proper solutions. Under suitable convexity assumptions, any weak solution of a vector equilibrium problem is a proper (hence strong) solution for at least one of its subproblems. Further generalizations may be obtained by considering more general classes of vector functions, following the approach proposed by Benoist, Borwein & Popovici (2003) for characterizing the cone-quasiconvex vector functions by means of extreme directions of the polar cone. 32 / 56

33 Proper efficiency Equilibrium conditions via scalarization Let Y R m be a nonempty set. Definition 19 A point y 0 = (y 0 1,..., y 0 m) Y is called properly minimal (Geoffrion, 1968) if there exists a real number µ > 0 such that for any y = (y 1,..., y m ) S and i {1,..., m} with y i < yi 0 j {1,..., m} \ {i} with µ(y j yj 0) y i 0 y i. there is Let PMinY be the set of properly minimal points of Y. We have PMin Y Min Y WMin Y. 33 / 56

34 Equilibrium conditions via scalarization According to Podinovskĭı & Nogin (1982), there exists a family (Λ ε ) ε ]0,1/m] of convex cones of R m, satisfying the properties: ε ]0,1/m] Λ ε = R m +; 0 m / int Λ ε, for any 0 < ε 1/m; Λ ε \ {0 m } int Λ ε, for any 0 < ε < ε 1/m; PMin Y = { } y 0 Y (Y y 0 ) ( Λ ε ) = {0 m }. 0<ε 1/m Note that Henig (1982) also introduced a notion of proper efficiency with respect to a general cone C, by means of certain enlarged cones with similar properties. Actually, for the particular cone C = R m +, several other concepts of proper efficiency coincide with Geoffrion s one (see, e.g., Guerraggio et al., 1994). 34 / 56

35 Equilibrium conditions via scalarization Theorem 20 (Hurwicz, 1958; Geoffrion, 1968) For any nonempty set Y R m the following hold: 1 argmin c, y PMin Y for every c R m with c > 0 m. y Y 2 If Y is convex (actually if Y is closely R m +-convex, i.e., cl(y + R m +) is convex), then WMin Y = PMin Y = argmin c 0 m y Y argmin c>0 m y Y c, y ; c, y. 35 / 56

36 Equilibrium conditions via scalarization Let D be a nonempty subset of a real linear space X. Assume that the bifunction ϕ : D D R satisfies the property that ϕ(x, x) = 0 for all x D. Recall that the scalar equilibrium problem governed by ϕ consists in finding the elements x D satisfying ϕ(x, y) 0 for all y D. 36 / 56

37 Equilibrium conditions via scalarization Due to the assumptions imposed on φ the set of all solutions of the scalar equilibrium problem becomes eq(d ϕ) := {x D ϕ(x, y) 0, y D} = {x D y D s.t. ϕ(x, y) < 0} = {x D ϕ(x, D) ( int R + ) = } = {x D 0 = min ϕ(x, D)}, where ϕ(x, D) := {ϕ(x, y) y D} and min ϕ(x, D) := min ϕ(x, y) for any x D. y D 37 / 56

38 Equilibrium conditions via scalarization Consider now a vector-valued bifunction, f = (f 1,..., f m ) : D D R m (m 2), which satisfies the property that f (x, x) = 0 m for all x D. The vector equilibrium problem governed by f consists in finding the elements of the following three sets: 38 / 56

39 Equilibrium conditions via scalarization w-eq(d f ) := {x D 0 m WMinf (x, D)} = {x D y D s.t. f (x, y) < 0 m } = { x D f (x, D) ( int R m +) = } ; s-eq(d f ) := {x D 0 m Minf (x, D)} = {x D y D s.t. f (x, y) 0 m } = { x D f (x, D) ( R m +) = {0 m } } ; p-eq(d f ) := {x D 0 m PMinf (x, D)} = {x D f (x, D) ( Λ ε ) = {0 m } for some ε ]0, 1/m]}. The elements of these sets are called weak solutions, strong solutions, and proper solutions of the vector equilibrium problem. 39 / 56

40 Equilibrium conditions Equilibrium conditions via scalarization Theorem 21 For any c R m let c, f : D D R be defined by c, f (x, y) := c, f (x, y). The following assertions hold: 1 eq(d c, f ) w-eq(d f ) whenever c 0 m. 2 eq(d c, f ) p-eq(d f ) whenever c > 0 m. 3 If the set f (x, D) is closely R m +-convex for every x D, then w-eq(d f ) = eq(d c, f ); c 0 m p-eq(d f ) = eq(d c, f ). c>0 m 40 / 56

41 Equilibrium conditions via scalarization For each nonempty set I I n we define an equilibrium subproblem as follows. If I = {i} is a singleton, then we consider the scalar equilibrium problem, whose solution set is eq(d f I ) := eq(d f i ). If I has cardinality I = k 2, then we consider the vector equilibrium problem, whose solution sets are w-eq(d f I ) := {x D 0 k WMinf I (x, D)}, s-eq(d f I ) := {x D 0 k Minf I (x, D)}, p-eq(d f I ) := {x D 0 k PMinf I (x, D)}. 41 / 56

42 Equilibrium conditions via scalarization Generalized convexity Definition 22 (Jeyakumar 1985; Breckner & Kassay 1997) A vector function g : D R m is called R m +-subconvexlike if there is e int R m + such that (1 t)g(d) + tg(d)+ ]0, [ e g(d) + R m +, t ]0, 1[. Actually, g is R m +-subconvexlike if and only if its range, g(d), is a closely R m +-convex set, i.e., the set cl(g(d) + R m +) is convex. Obviousy, if g is componentwise convex, then g(d) + R m + is a convex set, hence g is R m +-subconvexlike. 42 / 56

43 Equilibrium conditions via scalarization Theorem 23 If the bifunction f is R m +-subconvexlike (in particular, componentwise convex) in its second argument, then w-eq(d f ) = s-eq(d f I ) = =I I m =I I m p-eq(d f I ). 43 / 56

44 Equilibrium conditions via scalarization Given a vector-valued function, F : D R n, we can define a bifunction f : D D R n for all (x, y) D D by f (x, y) := F (y) F (x). In this case the vector equilibrium problem governed by f becomes a multi-criteria minimization problem. Thus, we recover and indeed refine the aforementioned result by Lowe et al. (1984). Similarly, by an appropriate choice of the bifunction f, certain types of vector variational inequalities can be also formulated as vector equilibrium problems. In particular, some recent results by Popovici & Rocca (2012, 2013) can be seen as counterparts of the above theorem. Further extensions of our results could be established by considering arcwise convexity, as in the paper by La Torre & Popovici (2010). 44 / 56

45 References I Equilibrium conditions via scalarization S. Alzorba, Chr. Günther, N. Popovici, A special class of extended multicriteria location problems, Optimization 64 (2015), Q. H. Ansari, and vector variational inequalities, In F. Giannessi (Ed.), Vector Variational Inequalities and Vector Equilibria. Mathematical Theories (pp. 1 16). Dordrecht, Boston, London: Kluwer Academic Publishers, J. Benoist, J.M. Borwein, N. Popovici, A characterization of quasiconvex vector-valued functions, Proc. Amer. Math. Soc. 131 (2003) / 56

46 References II Equilibrium conditions via scalarization J. Benoist, N. Popovici, The structure of the efficient frontier of finite-dimensional completely shaded sets, J. Math. Anal. Appl. 250 (2000), J. Benoist, N. Popovici, Contractibility of the efficient frontier of three-dimensional simply-shaded sets, J. Optim. Theory Appl. 111 (2001), M. Bianchi, N. Hadjisavvas, S. Schaible, Vector equilibrium problems with generalized monotone bifunctions, Journal of Optimization Theory and Applications 92 (1997), / 56

47 References III Equilibrium conditions via scalarization E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63 (1994), J.-M. Bonnisseau, B. Cornet, Existence of equilibria when firms follows bounded losses pricing rules, Journal of Mathematical Economics 17 (1988) J.-M. Bonnisseau, B. Crettez, On the characterization of efficient production vectors. Econom. Theory 31 (2007) / 56

48 References IV Equilibrium conditions via scalarization A. Cambini & L. Martein, Generalized convexity and optimality conditions in scalar and vector optimization, In: N. Hadjisavvas, S. Komlósi and S. Schaible (eds.), Handbook of generalized convexity and generalized monotonicity, Nonconvex Optim. Appl. 76, Springer-Verlag, New York (2005), C. Carathéodory, Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rendiconti del Circolo Matematico di Palermo 32 (1911), / 56

49 References V Equilibrium conditions via scalarization E. U. Choo & D. R. Atkins, Bicriteria linear fractional programming, J. Optim. Theory Appl. 36 (1982), G. Debreu, Theory of Value, John Wiley, New-York, R. Durier, C. Michelot, Sets of efficient points in a normed space, J. Math. Anal. Appl. 117 (1986), L. Fejér, Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen, Mathematische Annalen 85 (1922), / 56

50 References VI Equilibrium conditions via scalarization F. Flores-Bazán, S. Laengle, G. Loyola, Characterizing the efficient points without closedness or free-disposability, Cent. Eur. J. Oper. Res. 21 (2013), A. M. Geoffrion, Proper efficiency and the theory of vector maximization, Journal of Mathematical Analysis and Applications 22 (1968), E. Helly, On sets of convex bodies with common points (in German), ahrbuch der Deutschen Mathematiker Vereinigung 32 (1923), / 56

51 References VII Equilibrium conditions via scalarization A. N. Iusem, W. Sosa, New existence results for equilibrium problems, Nonlinear Analysis: Theory, Methods & Applications 52 (2003), J. Jahn, Vector optimization. Theory, Applications, and Extensions, Springer, Berlin, V. Jeyakumar, Convexlike alternative theorems and mathematical programming, Optimization 16 (1985), T. J. Lowe, J.-F. Thisse, J. E. Ward, R. E. Wendell, On efficient solutions to multiple objective mathematical programs, Management Sci. 30 (1984), / 56

52 References VIII Equilibrium conditions via scalarization Lê D. Muu, W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Analysis: Theory, Methods & Applications 18 (1992), W. Oettli, A remark on vector-valued equilibria and generalized monotonicity, Acta Mathematica Vietnamica 22 (1997), J.-P. Penot, L optimisation à la Pareto: deux ou trois choses que je sais d elle, In: Structures Economiques et Econométrie, Lyon, 1978, Publication Math. Pau, 19 pp. (1978). 52 / 56

53 References IX Equilibrium conditions via scalarization V.V. Podinovskĭı, V.D. Nogin, Pareto Optimal Solutions of Multicriteria Optimization Problems (in Russian), Nauka, Moscow, N. Popovici, Pareto reducible multicriteria optimization problems, Optimization 54 (2005) N. Popovici, Structure of efficient sets in lexicographic quasiconvex multicriteria optimization, Operations Research Letters 34 (2006), N. Popovici, Involving the Helly number in Pareto reducibility, Operations Research Letters 36 (2008), / 56

54 References X Equilibrium conditions via scalarization N. Popovici, M. Rocca, M., Decomposition of generalized vector variational inequalities, Nonlinear Analysis: Theory, Methods & Applications 75 (2012), N. Popovici, M. Rocca, Scalarization and decomposition of vector variational inequalities governed by bifunctions, Optimization 62 (2013), N. Popovici, A decomposition approach to vector equilibrium problems, Ann. Oper. Res., Published online: 3 Apr / 56

55 References XI Equilibrium conditions via scalarization I. M. Stancu-Minasian, Fractional programming. Theory, methods and applications, Mathematics and its Applications 409, Kluwer, Dordrecht, / 56

56 Equilibrium conditions via scalarization Thank you for your attention! 56 / 56