SATOSHI SUZUKI Communicated by Daishi Kuroiwa (Received: October 6, 2011, Revised: December 28, 2011) Abstract. This paper is based on the author's thesis, "On duality theorems for quasiconvex programming". In this paper, we investigate duality theorems for quasiconvex programming as generalizations of results in convex programming, and consists of three topics. The first topic is about quasiconjugates and polar sets. The second is about three types of set containment characterizations. The third is about constraint qualifications for Lagrange-type duality theorem in quasiconvex programming.
1. I NTRODUCTION
Mathematical programming is the use of mathematical models in order to assist in taking decisions, and is one of the most powerful techniques for making optimal decisions. In the research of mathematical programming, duality theorems have been investigated by many researchers. Especially, in convex programming, du- ality theorems are very useful and powerful tool to find a solution. The Fenchel conjugate is one of the well known conjugate function for convex functions, and by using this conjugate, some types of set containment characterizations have been investigated. Recently, these set containment characterizations imply the weakest constraint qualification for Lagrange-type duality theorems which play important roles in convex programming problem. Quasiconvex functions is well known as a generalized notion of convex func- tions. Since the class of quasiconvex functions is wide, and include many func- tions which arise in mathematical programming problem in practice, quasiconvex programming can be applied to a lot of problems. In quasiconvex programming, various important theorems and notions were investigated, for example, quasi- conjugates [9, 16, 17, 30, 31], duality theorems [9, 30, 31], and optimality condi- tions [15, 31].  However, these results are not generalizations of results in convex 1 programming although quasiconvex programming is a generalization of convex pro- gramming. In this paper, we investigate duality theorems for quasiconvex programming as generalizations of results in convex programming, by using quasiconjugates in [9, 16, 17, 30, 31] and Penot and Volle's result in [15]. This paper consists of three topics. Section 2 deals with quasiconjugates and polar sets.  We introduce quasiconjugates in [9, 16, 17, 30, 31], and investigate mainly H -quasiconjugate.  Also, we introduce various types of polar sets which play important roles in set containment characterizations. Section 3 deals with three types of set containment characterizations. We in- vestigate the characterization by using H and R-quasiconjugates, λ-quasi and λ- semiconjugates, and the generator of quasiconvex functions, respectively.  Also, we introduce the notion of generator for quasiconvex functions, and show that set containment characterizations by using the generator is generalized results of set containment characterizations in convex programming. In the last of Section 3, we compare characterizations in this paper with the previous ones. Section 4 deals with newly constraint qualifications for Lagrange-type duality theorem in quasiconvex programming. These constraint qualifications is concerned with the set containment characterization by using the notion of generator. In Sec- tion 4.1, we investigated the closed cone constraint qualification for quasiconvex programming. We show that this constraint qualification is a generalized notion of constraint qualification in convex programming and the weakest constraint quali- fication of Lagrange-type (strong) duality theorem for quasiconvex programming. In Section 4.2, we introduce a new subdifferential for quasiconvex functions by using generator. Also, we investigate optimality conditions as generalizations of convex ones, and establish the weakest constraint qualification for these optimality conditions.
2. QUASICONJUGATES AND POLAR SETS
It is well known that the Fenchel conjugate provides dual problems of convex pro- gramming problems. In a similar way, different notions of conjugate for quasiconvex functions have been introduced in order to obtain dual problems of quasiconvex programming problems.  For example, the λ-quasiconjugate (λ ∈ R), defined by Greenberg and Pierskalla [9], plays an important role in quasiconvex programming and in the theory of surrogate duality, corresponding to that of the Fenchel conju- gate in convex programming and Lagrangian duality.  But λ -quasiconjugate involves an extra parameter that many authors have tried to eliminate. Thach [30, 31] established two dualities without the extra parameter for a general quasiconvex minimization (maximization) problem, by using the concepts of H -quasiconjugate and R-quasiconjugate, which are similar to 1 and −1-quasiconjugate. In this section, we investigate some quasiconjugates and polar sets. All results of this section is based on [20, 29]. 2.1. H -quasiconjugate and R -quasiconjugate. First of all, we introduce the H -quasiconjugate of functions. In [30], Thach defined the H -quasiconjugate in R n , and in this section, we investigate the H -quasiconjugate also in R n . Definition 2.1. [30] Let f be a function from R n to R. The H -quasiconjugate of f is the function fH : Rn → R such that
The H -quasiconjugate of f H , denoted by fHH , is called the H -biquasiconjugate of f on R n \ {0} when f ≥ g on R n \ {0}, and f H = g H on R n \ {0}, f H ≥ g H Rn \ {0}. Also we have the following inequalities. Proposition 2.2. The following inequalities hold: (i) sup f H (u) ≤ − inf f (x), (ii) - sup f(x) < uERn\{0} reRn We can check that f H is H -evenly quasiconvex in a similar way of [31]. We can also see that the equality fHH(0) = inf{f(x) | x ∈ Rn \ {0}} holds in Lemma 4.1 of [30]. From this equality, we characterize the identity f = f HH in the next theorem. Theorem 2.3. The following properties are satisfied: (i) f = fHH on Rn \ {0} if f is H-evenly quasiconvex, (ii) f = fHH if and only if f is H-evenly quasiconvex and f(0) = inf{f(x) | x ∈ R n \ {0}}. Proof. At first, we show that (i) holds. It is clear that f(x) ≥ fHH(x) for all x ∈ R n \ {0}. Assume that there exists x 0 ∈ R n \ {0} such that f(x 0 ) > fHH(x 0 ). We can choose α ∈ R satisfying f(xo) > α > fHH(xo), and then x 0 ξ L ( f, ≤,α). Since L ( f, ≤,α) is H -evenly convex, there exists v ∈ R" \ {0} such that 〈v,xo〉≥_1 > 〈v,y〉for all y ∈ L(f,≤,α) by using separation theorem. This shows that fH (v) = − inf{f(x) | (v,x) ≥ 1} ≤ −α. Hence, fHH (x 0) = − inf { fH (u) | ⟨ u, x 0⟩ ≥ 1} ≥ − fH (v) ≥ α, and this is a contradiction. Next, we show that (ii) holds. By Lemma 4.1 of [30], fHH(0) = inf{f(x) | x ∈ R n \ {0}}. Hence, by using (i), we can prove that if f is H -evenly quasiconvex and f(0) = inf{f(x) | x ∈ R n \ {0}}, then f = f HH . The converse is clear since f HH is ■ H-evenly quasiconvex and fHH (0) = inf { f ( x ) | x ∈ R n \ {0}}. A function f is said to achieve the maximum value at infinity if f(xk) → sup{f(x) | x ∈ R n } for any sequence { x k } with || x k || → +∞, and f is said to achieve the minimum value at the origin if f(xk) → inf{f(x) | x ∈ R n \ {0}} for any sequence { x k } ⊂ R n \ {0} with x k → 0.  Let Γ∞ and γ 0 be the set of all functions that achieve the maximum value at infinity, and the set of all functions that achieve the minimum value at the origin, respectively, that is, Г∞ = {g : R n → R | g achieves the maximum value at infinity}, {g : R n → R | g achieves the minimum value at the origin}. γ 0 = We denote by S c the complement of S ⊂ R n and by B ( z, r ) the open ball centered at z E R n with radius r > 0. Proposition 2.4. The following properties are satisfied, (i) f ∈ Γ∞ if and only if for any M < sup{f(x) | x ∈ Rn} there exists δ > 0 such that B(0, δ)ς ⊂ L(ƒ, ≥, M), (ii) ƒ ∈ γ 0 if and only if for any m > inf{f(x) | x ∈ R n \ {0}} there exists δ > 0 such that B(0, δ) \ {0} ⊂ L(f, <, m). Proof. We only show (ii), and we can show (i) in the similar way. Assume that f achieves the minimum value at the origin and there exists m 0 > inf{f(x) | x ∈ R n \ {0}} such that for any δ > 0, there exists x ∈ B(0, δ) \ {0} such that f(x) ≥ m 0 . Then we can choose a sequence {x k} ⊂ R n \ {0} which converges to 0 and f( x k ) ≥ m 0 for each k ∈ N . This contradicts that f achieves the minimum value at the origin. Conversely, assume that for any m > inf{f(x) | x ∈ R n \ {0}}, there exists δ > 0 such that B(0, δ)\{0} ⊂ L(f, <, m). If { x k } ⊂ R n\{0} converges to 0, then there exists K ∈ N such that ‖ x k ‖ < δ for any k ≥ K . This shows that inf{f(x) | x ∈ R n \ {0}} ≤ f(x k) < m for any k ≥ K . This shows that {f(x k)} converges to inf{f(x) | x ∈ R n \ {0}}. According to [30], f is usc then fH is lsc, and if a function f ∈ Γ∞ is lsc, then fH is usc. Also, we investigate the following theorem. Theorem 2.5. The following properties are satisfied: (i) If f ∈ γ 0 then fH ∈ L∞ (ii) if f ∈ L∞ then fH ∈ γ⁰. Proof. (i) Let f ∈ γ 0 and { u k } ⊂ R n be a sequence satisfying ∥ u k ∥ tends to +∞. By using (ii) of Proposition 2.4, for any m > inf{f(x) | x ∈ R n \ {0}}, there exists δ > 0 such that B(0, δ) \ {0} ⊂ L(f, <, m). Since ‖u k ‖ tends to +∞ , we can find an integer K such that for any k ≥ K , ∥ u k ∥ z ∈ B(0, δ) \ {0}. By using (i) of Proposition 2.2, we can show that
because { ′ Uk, µµµ² = 1. This shows that { f H ( uk )} converges to sup{ f H ( u ) | u ∈ Rn}, and then fH ∈ L∞. (ii) Let f ∈ Γ∞ and { u k } ⊂ ℝ n \ {0} be a sequence satisfying { u k } converges to 0. For any M < sup{f(x) | x ∈ R n }, there exists δ > 0 such that B(0, δ)ς ⊂ L(ƒ, ≥, M), by Proposition 2.4 (i). Since { u k } converges to 0, we can find an integer K such that for any k ≥ K, {x | {uk, x} ≥ 1} ⊂ B(0, δ)ς, that is, ⟨ u k , x ⟩ ≥ 1 ⇒⇒ f ( x ) ≥ M . From this implication and by using (ii) of Proposition 2.2, we have sup{f(x) | x ∈ R n } ≥ −   inf    fH(u) ≥ − fH(uk) ≥ M, for any k ≥ K . This shows that { f H ( u k )} converges to inf { f H ( u ) | u ∈ R n \ {0}}, and then fH E γ°. ■ Next we show properties of level sets of H -biquasiconjugate. For this purpose, we prove the following proposition. Proposition 2.6. Let α, β ∈ R, and ν ∈ R n \ {0}. If f ∈ Γ∞ is lsc, then the following two conditions are equivalent: (i) L(f, ≤, β) ⊂ { x | { v, x } < α }, (ii) ∃ε > 0 s.t. L(f,<,β+ε) ⊂ {x | ⟨v,x⟩ < α}. Proof. We show that condition (i) implies condition (ii). Assume that L(f, ≤, β) ⊂ {x | ⟨ v,x ⟩ < α } and for all ε > 0 there exists x ε ∈ L(f,<,β + ε ) such that (v, x ε ) ≥ α , then we can choose a sequence { x k } ⊂ R n such that for all k ∈ N, β < f(xk) < β + ξ and ⟨ v, xk⟩ ≥ α, and we have f(xk) → β < f(x 1 ) ≤ sup{f(x) | x ∈ R n }. If ∥ x k ∥ → +∞, then f ( x k ) → sup{ f ( x ) | x ∈ R n } since f ∈ Γ∞ and this is a contradiction. If { x k } is bounded, then we can choose a subsequence { x k k } and x 0 ∈ R n such that x k i → x 0 . Clearly (v,x 0 ) ≥ α, but x 0 ∈ L(f,≤,β) since f ( x 0 ) ≤ lim inf i →∞ f ( x k i ) = β . This is a contradiction. The converse implication ■ is obvious. Now we can give results on the level sets of the H -biquasiconjugate. Theorem 2.7. The following properties are satisfied: (i) L(f, ≤,α) \ {0} ⊂ L(fHH, ≤,α), (ii) L(f,<,α) \ {0} ⊂ L(f,H,<,α), (iii) if α ≥ inf{f(x) | x ∈ R n \ {0}}, then HecL(f, ≤,α) ⊂ L(fHH, ≤,α), (iv)  L( fHH , <, α) ⊂ HecL( f, <, α),
Proof. (i), (ii) and (iii) are obvious. At first, we show (iv). Assume that x(≠ 0) f HecL(f,<,α). By using separation theorem, there exists a ∈ R n \ {0} such that ⟨a, x⟩ ≥ 1 > ⟨a, y⟩ for all y ∈ L( f, <, α). Then ƒнн ( x ) = − inf{ƒ H ( v ) | ⟨ v, x ⟩ ≥ 1} ≥ −ƒ H ( a ) = inf{ƒ( y ) | ⟨ a, y ⟩ ≥ 1} ≥ α. Therefore x f L(fHH,<,α). If L(fHH,<,α) contains 0, then L(f,<,α) is not empty since α > fHH(0) = inf{f(x) | x ∈ Rn \ {0}}. Hence HecL(f,<,α) contains 0. Next we show (v). By using (iii) and (iv), we can check easily that
We assume that x f HecL(f,<,α). By using separation theorem, there exists a ∈ R n \ {0} such that ⟨ a, x ⟩ ≥ 1 > ⟨ a, y ⟩ for all y ∈ L( f, ≤, α ), that is, L( f, ≤ ,α) ⊂ {y | ⟨ a,y ⟩ < 1}. By using Proposition 2.6, there exists ε 0 > 0 such that ⟨ a, x ⟩ ≥ 1 > ⟨ a, y ⟩ for all y ∈ L ( f, <, α + ε 0 ). By using separation theorem again, we have x̸= ∩ ε > 0 HecL( f, <, α + ε ), and consequently
This completes the proof.
By the definition of H -quasiconjugate and Theorem 2.3, we can see (i) (inf i ∈ I f i ) H = (sup i ∈ I f i H ) on R n \ {0}, and (ii) If f is H-evenly quasiconvex, then fHH = f on R n \ {0}. When every f i is H-evenly quasiconvex, by substituting f H into (i) we have
on R² \ {0}. However, the H -evenly quasiconvexity assumption is too strong The assump- because it assures fi(0) ≤ fi(x) for all x ∈ Rn and i ∈ I. tion of the next theorem is weaker than the previous one and guarantees that (inf i ∈ 1 f i ) HH = (sup i ∈ 1 f i ) H on R n \ {0}. Theorem 2.8. Let I be an arbitrary index set, and fi be an evenly quasiconvex function from R n to R for each i ∈ I. If the condition sup f i (x) > sup f i (0) for all x ∈ R n \ {0}, (A1) iEI iEl is satisfied, then
Proof. In general, the following equality about H -quasiconjugate of inf-function is satisfied: for all v ∈ R n \ {0},
hence, for all v ∈ R n \ {0}, we have
If the equality does not hold in the above inequality, then there exists v ∈ R n \ {0} such that (sup f i ) H ( v ) < (inf f i ) HH ( v ), and hence, there exists α ∈ R and x ′ ∈ R n such that 〈v, x′〉≥ 1 and
From x'̸= 0 and the assumption, we have sup i ∈ I f i ( x' ) > sup i ∈ I f i (0), and put ε′ = (sup i ∈ I f i ( x ′ ) − sup i ∈ I f i (0))/2 > 0. For all ε ∈ (0 ε ′ ), there exists i 0 ∈ I such that
Since L( f i0 , ≤ , sup i ∈ I f i ( x ′ ) − ε ) does not contain x ′ , contains 0, and it is evenly convex, there exists a ∈ R n \ {0} such that for all x ∈ L(f io , ≤, sup i ∈ I f i ( x ′ ) − ε ),
Therefore,
Since ε is arbitrary, we have
This is a contradiction.
Next, we introduce the R-quasiconjugate. Definition 2.9. [31] The R -quasiconjugate of f is the function f R : X → R such that f R (u) = − inf { f ( x ) | ⟨ u, x ⟩ ≥ − 1}, ∀ u ∈ X . Thach [31] investigated the R -quasiconjugate and introduced the following du- ality theorem. Let A be a nonempty convex subset of R n , and f is a quasiconvex function from Rn to R. Consider the following problem (P) and (D)
subject to u ∈ {v ∈ Rn | ∀ x ∈ A, ⟨v, x⟩ ≤ 1}. Let v(P) = inf{f(x) | x ∈ A} and v(D) = inf{f R (u) | ∀ x ∈ A, ⟨ u,x ⟩ ≤ 1}, then v(P) ≥ −v(D). Thach investigated necessary and sufficient conditions for ν(P) = −ν(D). Also, Thach introduced some optimality conditions by using R -quasiconjugate. 2.2. λ-quasiconjugate and λ-semiconjugate. Next, we introduce two concepts of quasiconjugate for quasiconvex functions. In [9], the following λ-quasiconjugate was investigated in R n . In this thesis, we define the λ -quasiconjugates in a locally convex topological vector space X . Definition 2.10. [9] The λ-quasiconjugate of f is the function f ζ : X* → R such that
Definition 2.11. [16] The λ-semiconjugate of f is the function f λ : X* → R such that
Singer [16] defined the λ-semiconjugate in the following form, f χ ( u ) = λ − 1 − inf { f ( x ) | ⟨ u, x ⟩ > λ − 1}, ∀ u ∈ X *. But we redefine the λ-semiconjugate in this paper.  Clearly, f i = f H + 1 and f ζ 1 = f R − 1. Also, we can check that f ζ is H -evenly quasiconvex and f ζ is lsc H -evenly quasiconvex if λ > 0 in the similar way of [16, 31]. Also, in [9, 16, 17, 31], researchers investigated the condition which guarantees that a function is equal to its biconjugate function and duality theorems by these quasiconjugates. 2.3. Polar sets. In this section, we introduce some types of polar sets in locally convex topological vector spaces. Definition 2.12. Let A be a nonempty subset of X and α ∈ R. We define polar sets as follows.
Also, S be a nonempty subset of X* and α ∈ R , we define polar sets as follows. S*(<,α) = { x ∈ X | ∀v ∈ S, { v, x } < α}, and S*(<,α) = { x ∈ X | ∀v ∈ S, { v, x } ≤ α}. Clearly, if α > 0 then A*(<,α) is H-evenly convex, ( A*(<,α))*(<,α) is HecA, A*(<,α) is closed H -evenly convex, and ( A *(≤ ,α))*(≤,α) is cl HecA. Moreover, for all α ∈ R,
Also, we introduce following two propositions. Proposition 2.13. Let I be an arbitrary set, A, be a nonempty H-evenly convex subset of X  for each i ∈ I , and α > 0.
Proposition 2.14. Let A be a nonempty subset of X and α ∈ R. Then, following statements hold. (i) (clcoA)*(≤,α) = A*(≤,α) and (ecA)*(<,α) = A*(<,α) (ii) if α > 0 then (cl Hec A )*(<,α) = A*(<,α) and (Hec A )*(<,α) = A*(<,α) Proof. At first, we show the statement (i). By separation theorem, for all v ∈ X* and α ∈ R, A ⊂ { x ∈ X | ⟨ v, x ⟩ ≤ α } if and only if cl co A ⊂ { x ∈ X | ⟨ v, x ⟩ ≤ α } . Also, by separation theorem, A ⊂ { x ∈ X | ⟨ v, x ⟩ < α } if and only if ec A ⊂ { x ∈ X | 〈v,x〉< α}, this completes the proof of (i). The proof of (ii) is similar by separation theorem.
3. SET CONTAINMENT CHARACTERIZATION
Classification is one of the basic problems in data mining which addresses the question of how best to use historical data to improve the process of making deci- sions and to discover regularities. Motivated by general nonpolyhedral knowledge- based data classification, the containment problem which consists of characterizing the inclusion A ⊂ B , where
and
was studied by many researchers [6, 8, 10, 14]. The first characterizations were given by Mangasarian [14] for linear systems and for systems involving differentiable convex functions, and keys to this approach were Farkas' Lemma and the duality theorems of convex programming, respectively.  In [10], Jeyakumar proved the following set containment characterizations. Theorem 3.1. [10] Let I be an arbitrary set, and for each i ∈ I, let f i be a convex function from R n to R. In addition, let { x ∈ R n | ∀i ∈ I, f i ( x ) ≤ 0} be a non-empty set, u ∈ R n , α ∈ R. Then, (i) and (ii) given below are equivalent: (i) {x ∈ R n | ∀i ∈ I, f i (x) ≤ 0} ⊂ {x ∈ R n | ⟨ u, x ⟩ ≤ α }, (ii) ( u, α) ∈ cl cone co |   | epif,* iEI Theorem 3.2. [10] Let I be an arbitrary set, and for each i ∈ I, let f, be a convex function from R n to R, {x ∈ R n | ∀ i ∈ I,f i (x) ≤ 0} be a non-empty set. In addition, let h be a convex function from R n to R. Then, (i) and (ii) given below are equivalent: (i) {x ∈ R n | ∀i ∈ I, f i (x) ≤ 0} ⊂ {x ∈ R n | h(x) ≥ 0}, (ii) (0,0) ∈ epih* + cl cone co iEI Jeyakumar investigated Theorem 3.1 and 3.2 in R n . However, these results are also valid in locally convex topological vector space because these results were proved by separation theorem. Also, these characterizations are generalization of Mangasarian's characterization, and play important roles in convex programming problems. In this section, we investigate three types of set containment characterizations for quasiconvex programming, by using H and R-quasiconjugates, λ-quasi and λ-semiconjugates, and the generator of quasiconvex functions, respectively. This section is based on [20, 29, 24]. 3.1.  By H -quasiconjugate and R -quasiconjugate.  In this section, we establish dual characterizations of the set containment in R n , assuming the quasiconvexity of g j , j ∈ J , the linearity or quasiconcavity of k s , s ∈ S , that A is defined by strict inequalities and B by both types of inequalities, so that A is convex whereas B is either convex or reverse convex. The dual characterizations are provided in terms of level sets of H -quasiconjugate and R -quasiconjugate of quasiconvex functions. We present a characterization of the containment of a convex set, defined by quasiconvex constraints, in an open halfspace. We start with a result on the con- tainment for the case | J | = 1. Theorem 3.3. Let v ∈ R n \ {0}, α ∈ (0, ∞) and β ∈ R.  Then the following conditions are equivalent:
Proof. Assume that L(g,<,β) ⊂ { x | { v,x } < α }, that is, g(x) < β implies {v,x} < α or, equivalently, { ξ,x} ≥ 1 implies g(x) ≥ β . This shows that
Therefore the Conversely, if gH(ﾆ) ≤ −β , then inf{g(x) | 〈ﾆ,x ⟩ ≥ 1} ≥ β . inequality 〈 ∞ , x 〉 ≥ 1 implies g ( x ) ≥ β , that is, g ( x ) < β implies 〈 v, x 〉 < α . Thus L(g, <, β) ⊂ { x | { v, x } < α}. ■ The theorem is valid when the constraint function is unique. Substituting supj∈J 9j into g, we have
for v ∈ R n \ {0}, α ∈ (0, ∞) and β ∈ R. Combining Theorems 2.8 and 3.3, we get the first characterization theorem. Theorem 3.4. Let J be an arbitrary index set, and g j be an evenly quasiconvex function from R n to R for each j ∈ J, and assume the condition (A1): ∀ェ∈ R² \ {0} (A1) sup g j (x) > sup g j (0). jEJ JEJ Then, for v ∈ R n \ {0}, α ∈ (0,00) and β ∈ R, the following conditions (i) and (ii) are equivalent: (i) {x ∈ R n | sup j ∈ J 9 j (x) < β } ⊂ {x ∈ R n | {v,x} < α}, ∈ L((inf g H ) HH , ≥ , −β ). From Theorem 2.7 (iii) and Theorem 2.8,
which yields the following result. Corollary 3.5. Let v ∈ R n \ {0}, α ∈ (0, ∞) and β ∈ R. If there exists m ∈ N, v1,...,vm ∈ Rn, and λ 1,...,λm ∈ [0,00) with ∑ k =1 λ k ≤ 1 such that
then,
Next, we show a result on the containment for the case J is arbitrary. Theorem 3.6. Let J be an arbitrary index set, and g j be an evenly quasiconvex function from R n to R for each j ∈ J. Assume that the following conditions (A1) and (A2) are satisfied: (A1) ∀ x ∈ R n \ {0}
(A2) inf j ∈ J (g H ) is l.s.c and included in Γ ∞ . Then for v ∈ Rn \ {0}, α ∈ (0,∞) and β ∈ R, the following statements are equivalent: (i) L(sup g j , <, β ) ⊂ { x ∈ R n | { v, x } < α }, (ii) = ∈ HecL(inf, g f , ≤ , −β). Proof. Firstly, we show that (ii) implies (i). Assume that (ii) holds. Then, from Theorem 2.7 (iii) and Theorem 2.8, HecL(inf g j , ≤, −β) ⊂ L((inf(g j )) HH , ≤, −β) = L((sup g j ) H , ≤, −β). Then we have  , ∈ L((sup i ∈ J g j ) H , ≤ , − β ) and, by Theorem 3.3, (i) is derived. Next, we show (i) implies (ii). By using Theorem 2.8 and Theorem 3.3, (i) implies
From assumption (A2) and Theorem 2.7, we get L((inf(g) , )) HH , ≤ , −β) = HecL(inf g) , ≤ , −β). Then (ii) is satisfied.
In the following Corollary 3.7, we show a set containment characterization when all g j are usc quasiconvex, J is a finite set, all k s are affine, S is an arbitrary set. Corollary 3.7. Let J be a finite set, S be an arbitrary set, gj be an usc quasiconvex function from R n to R and included in γ 0 for each j ∈ J, v s ∈ R n \ {0} and α s ∈ (0, ∞) for each s ∈  S . If condition (A1) holds, then the following conditions (i) and (ii) are equivalent: (i) {x ∈ R n | ∀ j ∈ J, g j (x) < β } ⊂ {x ∈ R n | ∀ s ∈ S, {v s,x} < α s }, ν 8 ∈ Hec (  ) L ( g 3 , ≤, −β ). (ii) ∀ s ∈ S, α ε jEJ Proof. We can check that U j ∈ J L(g f , ≤ , −β) = L(inf j ∈ J g f , ≤ , −β), L(sup j ∈ J , < , β ) = { x ∈ R n | ∀ j ∈ J, g j ( x ) < β } and (A2) by using assumptions. We present a characterization of the containment of a convex set, defined by finitely many constraints, in a reverse convex set, defined by a quasiconvex con- straint. Theorem 3.8. Let v ∈ R n \ {0}, β ∈ R. Then the following conditions are equivalent: (i) L(g,<,β) ⊂ {x | ⟨v,x⟩ < −1}, (ii) ν ∈ L(g R , ≤, −β). The proof is similar to the one of Theorem 3.3, hence it is omitted. Theorem 3.9. Let g and h be usc quasiconvex functions from R n to R. Assume that L(h,<,α)̸= ∅ and 0 ∈ L(g,<,β) for some α,β ∈ R. Then the following conditions are equivalent:
(iii) there exists v ∈ R n \ {0} such that g H (v) ≤ −β and h R (−v) ≤ −α. Proof. It is clear that (ii) and (iii) are equivalent. Since L(g,<,β) and L(h,<,α) are nonempty open convex subsets and 0 ∈ L(g, <,β), we have
by separation theorem.
Substituting sup j ∈ J 9j into g , we obtain the following theorem. Theorem 3.10. Let J be a finite set, gj be an usc quasiconvex function from R n to R for each j ∈ J, and h be an usc quasiconvex function from R n to R. Assume that L(h,<,α)̸= ∅ and sup j ∈ J 9j (0) < β for some α, β ∈ R. Then the following conditions are equivalent:
(iii) there exists v ∈ R n \ {0} such that (sup g j ) H ( v ) ≤ −β and h R (− v ) ≤ −α. JEJ Proof.  Since J is a finite set, we have sup j ∈ J g j is usc and L (sup j ∈ J g j , ≤ , β ) = { x ∈ Rn | ∀j ∈ J, gj (x) < β } . By using Theorem 3.9, we conclude the proof. The next corollary characterizes the set containment in the case that all g j are usc quasiconvex, J is a finite set, all h w are usc quasiconvex, W is an arbitrary set. Corollary 3.11. Let J be a finite set, W be an arbitrary set, g, be an usc quasicon- vex function from R n to R included in γ 0 for each j ∈ J, h w be an usc quasiconvex function from R n to R and α w ∈ (0, ∞) for each w ∈ W . Assume that (A1) ∀ x ∈ R n \ {0} sup g j (x) > sup g j (0), jEJ and L(h w , <,α w )̸= θ for each j ∈ J and sup i ∈ I f i (0) < β for some β ∈ R. Then the following conditions are equivalent: (i) {x ∈ R n | ∀ j ∈ J, g j (x) < β } ⊂ {x ∈ R n | ∀ w ∈ W, h w (x) ≥ α w }, (ii) for each w ∈ W , 0 ∈ Hec U j ∈ J L(g H , ≤ , −β) \ {0} + L(h R , ≤ , −α w ) \ {0},
3.2. By λ-quasiconjugate and λ-semiconjugate. In this section, we show set containment characterizations, by λ -quasiconjugate and λ -semiconjugate. These dual characterizations are provided in terms of level sets of λ -quasiconjugate and λ-semiconjugate, especially 1, −1-quasiconjugate, 1-semiconjugate.  In [29], we investigated characterizations in R n . However, in this section we investigate char- acterizations in a locally convex topological vector space. At first, we introduce the following proposition in a locally convex topological vector space X. Proposition 3.12. Let A be a convex subset of X, x ∈ clA and y ∈ intA. Then, for all α ∈ (0,1), (1 − α) x + αy ∈ int A . Proof. Since y ∈ int A , there exists a convex and symmetric neighborhood U of 0 such that y + U ⊂ A. Then, (1 − α) x + αy + α²U ⊂ A. Actually, since x ∈ clA and U is symmetric, there exists z ∈ αU such that z + x ∈ A and − z ∈ αU . For all a E α²U,
&
This completes the proof. The following proposition, which concerns the closure of the intersection of a family of convex sets, plays an important role in set containment characterizations. Proposition 3.13. Let I be an arbitrary set, and Ai be a convex subset of X for each i ∈ I .  If int ∩ i ∈ I A i  is nonempty, then cl ∩ i ∈ I A i  = ∩ i ∈ I Proof. Let x ∈ ∩ i ∈ I cl A i . Since int ∩ i ∈ I A i̸= ∅ , there exists z ∈ int ∩ i ∈ I A i . Then for each i ∈ I , {(1 − α) x + αz | α ∈ (0, 1]} ⊂ int A i , because of Proposition 3.12. Therefore {(1 − α)x + αz | α ∈ (0, 1]} ⊂ ∩ i ∈ I A i , that is, x ∈ cl ∩ i ∈ I A i . The converse is clear. We present characterizations of the containment of a convex set, defined by infinite quasiconvex constraints, in an evenly convex set. In the beginning, we show a result of the containment when | J | = | S | = 1. Theorem 3.14. Let g be a function from X to R, v ∈ X*\{0}, α ∈ R and β ∈ R. Then, following conditions (i), (ii) and (iii) are equivalent: (i) L(g, <, β) ⊂ { x | { v, x } < α }, (ii) ν ∈ (L(g, <, β))*(<,α), (iii) ν ∈ L(g ζ, ≤, α − β). Proof. It is clear that (i) and (ii) are equivalent. Assume that L(g,<,β) ⊂ {x | {v,x} < α}, then the implication g(x) < β implies {v,x} < α, or equivalently, 〈v, x〉≥ α implies g(x) ≥ β holds. This shows g'a(v) = α − inf{g(x) | {v,x} ≥ α} ≤ α − β. Conversely, if g ( v ) ≤ α − β , then inf { g ( x ) | { v, x } ≥ α } ≥ β . Hence, the implica- tion ⟨ v, x ⟩ ≥ α implies g ( x ) ≥ β, or g ( x ) < β implies ⟨ v, x ⟩ < α holds. This derives L(g,<,β) ⊂ {x | {v,x} < α}. Next, we show the set containment characterization, assuming that all g i ( j ∈ J ) are strictly evenly quasiconvex, J and S are possibly infinite. Theorem 3.15. Let J, S be arbitrary sets, β ∈ R, gj be a strictly evenly quasicon- vex function from X to R for each j ∈ J, and vs ∈ X* and α s ∈ (0, ∞) for each s ∈ S. Assume that g i (0) < β for each j ∈ J. Then, following conditions (i) and (ii) are equivalent: (i) {x ∈ X | ∀ j ∈ J, g j (x) < β } ⊂ {x ∈ X | ∀ s ∈ S, {v s, x) < α s }, (ii) ∀ s ∈ S , ν 8 ∈ Hec ( J L (( g 3 ) 5 , ≤ , 1 − β ). α ε jEJ Proof. It is clear that (i) and
are equivalent. By using the assumption, L ( g j , < , β ) is a H -evenly convex set for each j ∈ J . Hence, by using Proposition 2.13, ∂ ∈ Hec ∪ j ∈ J (L( g j , <, β ))*(<,1) for each s ∈ S . Furthermore, by using Theorem 3.14, c ∈ Hec U j ∈ L (( g j ) i , ≤ , 1 − β ) for each s ∈ S In the following theorem, we show the set containment characterization, assum- ing that all f i ( i ∈ I ) are evenly quasiconvex, I and S are possibly infinite. Theorem 3.16. Let I and S be arbitrary sets, β ∈ R, f i be an evenly quasiconvex function from X  to R  for each i ∈ I, and vs ∈ X* and α s ∈ (0,00) for each s ∈ S. Assume that f i (0) ≤ β for each i ∈ I . Then, following conditions (i) and (ii) are
Proof. It is clear that (i) and
are equivalent.  By using the assumption, L( fi, ≤, β ) is a H -evenly convex set for each i ∈ I.  Therefore, by using Proposition 2.13, for each s ∈ S, i ∈ E HecU i ∈ I ( L ( f i , ≤ , β ))*(<, 1 )   Because  L ( f i , ≤ , β ) =  ∩ ε > 0 L ( f i , ≤ , β + ε )   by  using Proposition 2.13 again, for each s ∈ S, ∞ ∈ Hec∪ i ∈ I Hec∪ ε> (L( f i, ≤, β+ε ))*(<,1) . prove that " ε HecU i ∈ I U ε > 0 L (( f i ) i , ≤ , 1 − β − ε ) for each s ∈ S by using Theo- rem 3.14. Therefore, for each s ∈ S, ∞ ∈ Hec ∪ i ∈ I L(( f i ) ζ , <, 1 − β ). The converse is similar. ■ Next, we show the set containment characterization, assuming that all f i ( i ∈ I ) are evenly quasiconvex, all g j ( j ∈ J ) are strictly evenly quasiconvex, I , J and S are arbitrary sets. Theorem 3.17. Let I, J and S be arbitrary sets, β ∈ R, fi be an evenly quasi- convex function from X to R for each i ∈ I, gj be a strictly evenly quasiconvex function from X  to R  for each j ∈ J, and v s ∈ X* and α s ∈ (0,00) for each s ∈ S. Assume that f i (0) ≤ β and g j (0) < β for each i ∈ I and j ∈ J. Then, following
Proof. The proof is similar to Theorem 3.15 and 3.16.
In the following theorem, we show the result of the characterizing set containment when |J| = |W| = 1 by using λ-semiconjugate. Theorem 3.18. Let g be a function from X to R, u ∈ X*, γ ∈ R and β ∈ R. Then, following conditions (i), (ii) and (iii) are equivalent: (i) L(g,<,β) ⊂ {x ∈ X | {u,x} ≤ γ}, (ii) u E (L(g, <, β))*(≤,γ) (iii)  u ∈ L(g 2 , ≤, γ − β). Proof. It is clear that (i) and (ii) are equivalent.  We may assume that L(g,< ,β) ⊂ { x ∈ X | ⟨ u, x ⟩ ≤ γ }, then the implication g( x ) < β implies ⟨ u, x ⟩ ≤ γ, or equivalently, { u, x } > γ implies g(x) ≥ β holds. This shows g γ (u) = γ − inf{g(x) | {u,x} > γ} ≤ γ − β. Conversely, if g f ( u ) ≤ γ − β , then inf{ g ( x ) | { u, x } > γ } ≥ β . Therefore the implication ⟨ u, x ⟩ > γ implies g( x ) ≥ β, or g( x ) < β implies ⟨ u, x ⟩ ≤ γ holds. This derives L(g, <, β) ⊂ { x | { u, x } ≤ γ}. Next, we show the set containment characterization, assuming that g i (j ∈ J) are quasiconvex, J and W are arbitrary sets. Theorem 3.19. Let J and W  be arbitrary sets, β ∈ R, g j  be a quasiconvex function from X to R for each j ∈ J, and u w ∈ X* and γ w ∈ (0,00) for each w ∈ W. Assume that gj(0) < β for each j ∈ J and int{x ∈ X | ∀j ∈ J,gj(x) < β} is nonempty. Then, following conditions (i) and (ii) are equivalent: QUASICONVEX DUALITY THEOREMS WITH QUASICONJUGATES AND GENERATOR 17
Proof. It is easy to show that (i) is equivalent to
convex for each j ∈ J . Hence, by using Proposition 2.13, for each w ∈ W , ⇌ ∈ cl Hec U j ∈ J (clL( g j , <, β ))*(≤ , 1)    Also, by using Theorem 3.18, for each w ∈ W , i = cl Hec U j ∈ J L (( g j ) i , ≤ , 1 − β ). The converse is similar. In the following theorem, we show the set containment characterization, assum- ing that all f i ( i ∈ I ) are quasiconvex, I and W are arbitrary sets. Theorem 3.20. Let I and W  be arbitrary sets, β ∈ R, fi be a quasiconvex function from X to R for each i ∈ I, and u w ∈ X* and γ w ∈ (0, ∞) for each w ∈ W. Assume that f i (0) ≤ β for each i ∈ I and int{x ∈ X | Vi ∈ I,f i (x) ≤ β} is nonempty. Then, following conditions (i) and (ii) are equivalent: (i) { x ∈ X | ∀i ∈ I, fi(x) ≤ β} ⊂ { x ∈ X | ∀w ∈ W, { u w, x } ≤ { w } u w ∈ cl Hec ( f i ) 1 , <, 1 − β ). (ii) ∀ w ∈ W , /w iEI Proof. It is clear that (i) is equivalent to for each w ∈ W, π x ∈ ( ∩ i ∈ I L ( f i , ≤ , β ))*(≤,1). By the similar way in Theorem 3.19, we can prove that for each w ∈ W , === ( { f i , <, β ))*(≤,¹) . From the assumption, clL( f i , ≤,β ) is a closed H- evenly convex set for each j ∈ J. Therefore, by using Proposition 2.13, W ∈ cl Hec U i ε l (clL( f i , ≤ , β))*(≤ 1 )  for  each  w  ∈  W .   Also,  by  using  Proposition  3.13
Furthermore,
and by using Theorem 3.18, we can prove that for each w ∈ W ,
Next, we show the set containment characterization, assuming that f i and g i are quasiconvex for each i ∈ I and j ∈ J , I , J and W are arbitrary sets. Theorem 3.21. Let I, J and W  be arbitrary sets, β ∈ R, f i and g j be quasiconvex functions from X to R for each i ∈ I and j ∈ J, and u w ∈ X* and γ w ∈ (0, ∞) for each w ∈ W . Assume that f i (0) ≤ β for each i ∈ I , g j (0) < β for each j ∈ J , and int{x ∈ X | ∀i ∈ I, f i (x) ≤ β,∀j ∈ J,g i (x) < β} is nonempty.  Then, following conditions (i) and (ii) are equivalent:
Proof. The proof is similar to Theorem 3.19 and 3.20. In the following theorem, we show the set containment characterization, assum- ing that all f i ( i ∈ I ) are evenly quasiconvex, all g j ( j ∈ J ) are strictly evenly quasiconvex, and I , J , S and W are arbitrary sets. Theorem 3.22. Let I, J, S and W be arbitrary sets, β ∈ R, f, be an evenly quasiconvex function from X to R for each i ∈ I , gj be a strictly evenly quasiconvex function from X to R for each j ∈ J, vs ∈ X* and α s ∈ (0, ∞) for each s ∈ S, and u w ∈ X* and γω ∈ (0,00) for each w ∈ W . Assume that f i (0) ≤ β for each i ∈ I, g i (0) < β for each j ∈ J and int{ x ∈ X | f i ( x ) ≤ β,i ∈ I, g i ( x ) < β,j ∈ J}
Proof. The proof is similar to Theorem 3.15, 3.16, 3.19 and 3.20. □ We present characterizations of the containment of a convex set, defined by infi- nite quasiconvex constraints, in a reverse convex set, defined by infinite quasiconvex constraints. In the beginning, we show the result of the containment when | J | = | W | = 1. Theorem 3.23. Let g be a quasiconvex function from X to R, h be an usc qua- siconvex function from X to R, γ ∈ R and β ∈ R. Assume that L(g,<,β) and L(h, <, γ) are nonempty. Then, following conditions (i), (ii) and (iii) are equiva- lent:
Proof. It is clear that (i) and (ii) are equivalent. We may assume that the condition (ii) holds. Then, there exists v ∈ X*\{0} and α ∈ R such that for all x ∈ L(h, <, γ) and y ∈ L(g, <, β), {v, x} > α ≥ {v, y} since g is quasiconvex and k is usc quasiconvex. Clearly, v ∈ (L(g, <,β))*(≤,α) and −ν ∈ (L(h, <, γ))*(<,−α). By using Theorem 3.14 and Theorem 3.18, ν ∈ (L(g δ, ≤ ,α − β) and −ν ∈ L(h −α, ≤, −α − γ). Hence, 0 ∈ L(g α, ≤,α − β) \ {0} + L(h −α, ≤ , −α − γ) \ {0}. The converse is similar. Next, we show the set containment characterization, assuming that all g j ( j ∈ J ) are quasiconvex, all h w ( w ∈ W ) are usc quasiconvex, J and W are arbitrary sets. Theorem 3.24. Let J and W  be arbitrary sets, gj be a quasiconvex function from X to R for each j ∈ J, h v be an usc quasiconvex function from X to R and Yw ∈ R for each w ∈ W, and β ∈ R.  Assume that 0 ∈ int ∩ j ∈ J L( g j , <, β ) and NwEW L(hw,<, γw) is nonempty.  Then, following conditions (i) and (ii) are equivalent:
Proof. Assume that the condition (i) is hold. Then, for each w ∈ W , ∩ i ∈ JL ( g i , < ,β) ∩ L( hw , <, γ w ) = ∅. Since all g j are quasiconvex, k w are usc quasiconvex and 0 ∈ int ∩ j ∈ J L ( g j , <, β ), there exists v ∈ X * \ {0} such that for all x ∈ L ( h w , < , γ w ) and y ∈ ∩ j ∈ JL (g j , <,β), ⟨ v, x ⟩ > 1 ≥ ⟨ v, y ⟩ . By using Theorem 3.14 and Theorem 3.19, we can prove that v ∈ cl Hec U j ∈ J L(( g j ) i , ≤ , 1 − β ) and − v ∈ L(( hw )ﾆ1 , ≤ , − 1 − γ w ), that is, 0 ∈ (cl Hec i j ∈ J L(( g j ) ↑ , ≤ , 1 − β ) \ { 0 } ) + L(( hw )ﾆ1 , ≤ , −1 − √w) \ {0}. The converse implication is similar. In the following theorem, we show the set containment characterization, assum- ing that f i ( i ∈ I ) are quasiconvex, h w ( w ∈ W ) are usc quasiconvex, I and W are arbitrary sets. Theorem 3.25. Let I and W  be arbitrary sets, fi be a quasiconvex function from X to R for each i ∈ I, h w be an usc quasiconvex function from X to R and Yw ∈ R for each w ∈ W , and β ∈ R. Assume that 0 ∈ int Ni∈ I L(fi,≤,β) and NwEWL(hw,<,√w) is nonempty.  Then, following conditions (i) and (ii) are
, γ w ) = ∅ for each w ∈ W . Since all f i are quasiconvex, h w are usc quasicon- vex and 0 ∈ int ∩ i ∈ I L ( f i , ≤ , β ), there exists ν ∈ X* \ {0} such that for all x ∈ L( h w , <, γ w ) and y ∈ ∩ i ∈ IL( f i , ≤ , β ), 〈 v , x 〉 > 1 ≥ 〈 v , y 〉 . By using Theo- rem 3.14 and Theorem 3.20, we can prove that v ∈ cl Hec Uj∈J L(( fi)1, <, 1 − β ) and − ν ∈ L(( h w ) − 1 , ≥ , − 1 − γ w ). The converse is similar. We show the set containment characterization, assuming that f i and g j are qua- siconvex for each i ∈ I and j ∈ J , h w are usc quasiconvex for each w ∈ W , and I , J and W are arbitrary sets. Theorem 3.26. Let I, J and W be arbitrary sets, fi and gj be quasiconvex func- tions from X to R for each i ∈ I and j ∈ J, hw be an usc quasiconvex func- tion from X to R and Yw ∈ R for each w ∈ W, and β ∈ R. Assume that 0 ∈ int[(↑ i ∈ IL( f i , ≤ , β)) ∩ (↑ i ∈ IL( g j , <, β))] and ∩ w ∈ W L( h w , <, √ w ) is nonempty.
In this section, we show set containment characterizations in an evenly convex set, assuming that the inequalities in A and B can be either weak or strict. However, on set containment characterizations in a reverse convex set, we show only the case where inequalities in B are weak. Hereinafter, we show that it is difficult to characterize the set containment characterization in a reverse convex set, assuming that the inequalities in B are strict. We consider the characterization of A ⊂ B , where I , J and S are arbitrary sets, f i and g j are quasiconvex functions from X to R for each i ∈ I and j ∈ J, k s is a quasiconvex function from X to R and α s ∈ R for each s ∈ S, β ∈ R, and A = {x ∈ X | fi(x) ≤ β,i ∈ I,gj(x) < β,j ∈ J}, B = {x ∈ X | k s (x) > α s , s ∈ S}. Assume that J is empty, f is lsc quasiconvex for each i ∈ I , k s is lsc quasiconvex and L( k s, ≤, α s ) is compact for each s ∈ S , and 0 ∈ int ∩ i ∈ I L( f i , ≤, β ). Then, following conditions (i) and (ii) are equivalent: (i)
(ii) ∀x ∈ L(k s, ≤, α s), ∀y ∈ ∩ i∈ I L(f i, ≤, β), 〈 v, x 〉 > 1 ≥ 〈 v, y 〉 . Of course, we can rewrite the condition (ii) by using level sets of quasiconjugate functions. Assume that I is empty, | J | < ∞ , g j is usc quasiconvex for each j ∈ J , k s is quasiconvex for each s ∈ S , and 0 ∈ int ∩ j ∈ J L ( g j , <, β ), then, following conditions (i) and (ii) are equivalent: (i)
(ii) ∀ x ∈ L(k s, ≤, α s), ∀ y ∈ ∩ j ∈ J L(g j , <, β), ⟨ v, x ⟩ ≥ 1 > ⟨ v, y ⟩ . Hence, we can show the set containment characterization by using 1-quasiconjugate and  —1-semiconjugate. However, if J is an arbitrary set, then ∩ j ∈ J L ( g j , <, β ) is not always open even if g j is usc quasiconvex for each j ∈ J . Hence, if g j is usc quasiconvex for each j ∈ J, k ｡ is lsc quasiconvex and L(k ｡ ≤ ,α ｡ ) is compact for each s ∈ S, 0 ∈ int ∩ j ∈ J L( g j , <, β ) and ∩ j ∈ J L( g j , <, β ) ⊂ ∩ s ∈ S L( k s , >, α s ), then, for each s ∈ S , there exists v ∈ X* \ {0} such that for all x ∈ ∩ j ∈ J L( g j , <, β ) and y ∈ L( k s , ≤, α s ),
Also, the above inequality does not imply that ∩ j ∈ J L( g j , <, β ) ⊂ ∩ s ∈ S L( k s , >, α s ), and these assumptions of functions are the strongest one in this problem.  Therefore, it is hard to characterize set containments by using quasiconjugate function. 3.3. Application. We show that set containment characterizations in this section is useful to consider quasiconvex programming problem. Let I be an arbitrary set, f i be a lsc quasiconvex function from X to R for each i ∈ I, A = {x ∈ X | Vi ∈ I, f i ( x ) ≤ 0}, and h be an usc quasiconvex function. Assume that 0 ∈ int A , and consider the following problem ( P ),
Remember that all lsc quasiconvex functions are the supremum of some family of lsc quasi-affine functions. Hence, without loss of generality, we can assume that f i is a lsc quasi-affine function for each i ∈ I , that is, there exist {( l i , v i ) | i ∈ I } ⊂ Q × X * such that f i = l i ov i for each i ∈ I . By using Theorem 3.25, for each γ ∈ R , following conditions (i), (ii) and (iii) are equivalent: (i) ∩ i∈I L( f i , ≤, 0) ⊂ L( h , ≥, γ ), (ii)  0 ∈ cl Hec U i ∈ I  L(( f i ) 1 , <, 1) + L( h − 1 , ≤ , − 1 − γ ) \ {0} (iii) 0 ∈ cl co( { π x = π σ ν i | i ∈ I} ∪ {0}) + L( k = 1 , ≤ , − 1 − γ ) \ {0}. Actually, for each i ∈ I,
If z f R+{vi}, it is clear that inf{li 0 vi(x) | {z,x} > 1} = inf ter li (t) ≤ 0 because S is nonempty.  And if z ∈ R+{vi} \ {0}, there exists λ > 0 such that z =  λν i , hence inf{l i o v i (x) | 〈z,x〉 > 1} = l i ( i ) because l i is non- decreasing.  Also, it is clear that inf{li o vi(x) | (0,x)  > 1} = ∞, hence 
L((l, o v,)l, <, 1) = clHec U,e, [0, [[]=[[]){v,}. Furthermore, clHec U,e, [0, [[]=[[]]{v,}.
L((l, o v,)l, <, 1) = clHec U,e, [0, [[]=[[]){v,} = clec(U,e,r[0, [[]=[[]){v,} because Uie1[0, [L]=1[W){vi} is nonempty.  Also clec(Uie1[0, [L]=1[W){vi} U {0}) = cl co(U i ∈ I [0, π ]= 1 σ ){ v i } ∪ { 0} ) = cl co({ π )= 1 σ v i | i ∈ 1 } ∪ { 0} ).  Hence, the above conditions (i), (ii) and (iii) are equivalent. Clearly, inf x∈ A h ( x ) = sup{ γ ∈ R | ∩ i∈ IL ( f i , ≤ , 0) ⊂ L ( h, ≥ , γ )} . Hence, we can prove that
−cl co({ π, )−−−(σ ν; | i ∈ I} ∪ {0}). Furthermore, A*(≤,ι) = cl co({ π,ι−−(σ ν; | i ∈ I}∪{0}) = −T and k −1 + 1 = h R . Hence,
and we can get another duality problem of ( P ),
This duality problem ( D ′ ) is a same problem of the duality problem by Thach [31]. 3.4.  By the generator of quasiconvex functions.  In this section, we show two set containment characterizations for quasiconvex constraints. First, we present a characterization of a containment of a convex set, defined by a quasiconvex con- straint, in a closed halfspace. First, we introduce a notion of a quasiaffine function which is a generalized notion of an affine function. A function f is said to be quasiaffine if quasiconvex and quasiconcave. In [15], Penot and Volle proved the following theorem. Theorem 3.27. [15] The following conditions (i) and (ii) are equivalent: (i) f is lsc quasiaffine, (ii) there exists k ∈ Q and w ∈ X* such that f = k ⊙ w, where Q = {h : R → R | h is lsc and non-decreasing}.  Also, the following condi- tions (iii) and (iv) are equivalent: (iii) f is lsc quasiconvex, (iv) there exists {( k i , w i ) | i ∈ I} ⊂ Q × X* such that f = sup i ∈ I k i ○ w i . Theorem 3.27 indicates that a lsc quasiconvex function f consists of a supremum of some family of lsc quasiaffine functions.  Based on this result, we define a notion of generator for quasiconvex functions, that is, G = {( k i , w i ) | i ∈ I } ⊂ Q × X * is said to be a generator of f if f = sup i ∈ I k i ○ w i , and we can see that all lsc quasiconvex functions have at least one generator by Theorem 3.27. Also, when f is a proper lsc convex function, Bf = {(kv, v) | v ∈ domf*, kv(t) = t − f*(v), ∀t ∈ R} ⊂ Q× X* is a generator of f . Actually, for all x ∈ X, f(x) = f**(x) = sup{{v,x) − f*(v) | v ∈ domf*} =   sup   kv({v,x)} vEdom f* We call the generator Bf "the basic generator" of convex function f . The basic gen- erator is very important with respect to the comparison of convex and quasiconvex programming. Moreover, we introduce a generalized notion of inverse function of h ∈ Q . The following function h-1 is said to be the hypo-epi-inverse of h : h-¹(a) = inf{b ∈ R | a < h(b)} = sup{b ∈ R | h(b) ≤ a}. If h has the inverse function, then the inverse function of his equal to the hypo- epi-inverse of h. in detail see [15]. In this thesis, we denote the hypo-epi-inverse of h by h-1. Also, we denote the lower left-hand Dini derivative of h ∈ Q at t by D_h(t), that is D_h(t) = liminf ε→o_ h(t+ε)−h(t)   A function h is said to be lower left-hand Dini differentiable if D_h(t) is finite for all t ∈ R . By using the notion of generator, we investigate the following set containment characterization. Theorem 3.28. Let f be a lsc quasiconvex function from X to R with generator {(k i , w i ) | i ∈ I} ⊂ Q × X*, u ∈ X*, and α, β ∈ R. Assume that { x ∈ X | f(x) ≤ β}̸= θ and there exists i 0 ∈ I such that k i − 1 (β) ∈ R. Then (i) and (ii) given below are equivalent: (i) {x ∈ X | f(x) ≤ β} ⊂ {x ∈ X | ⟨u,x⟩ ≤ α}, (ii) (u,α) ∈ cl cone co   } {(w,,δ) ∈ X* × R | k,-1(β) ≤ δ}. iEI Proof. Let ( u, α ) ∈ cl cone co ∪ i ∈ 1 {( w i, δ ) ∈ X* × R | k − 1 ( β ) ≤ δ }, then there ex- ists {( u k , α k )} ⊂ coneco ∪ i ∈ I {( w i , δ ) ∈ X* × R | k j − 1 (β) ≤ δ } such that {( u k , α k )} converges to ( u,α). For each k ∈ N, there exist λ k ≥ 0, m k ∈ N, i 1, . . . ,imk ∈ I, δ 1 , · · , δ mk ∈ R, and β 1 , · · , β mk ≥ 0 such that Σ mk β j = 1, δ j ≥ k i − 1 (β), and ( uk , αk ) = λk (Σ j =1β j w i , Σ j =1β j δ j ). Then, for all x ∈ X with f(x) ≤ β and i ∈ I , k i ( \w i , x \) ≤ β . Hence, for all k ∈ N,
Therefore, ⟨ u, x ⟩ ≤ α. Conversely, let ( u, α )̸ξ cl cone co ∪ i ε I {( w,,δ) ∈ X* × R | k;-1(β) ≤ δ}.  By using separation theorem, there exists (v 0 , γ 0 ) ∈ (X × R)\{0} such that for all (w, δ) ∈ U i ∈ 1 {(w i, δ) ∈ X* × R | k − 1 (β) ≤ δ}, ((u, α), (vo, γο)) > 0 ≥ 〈(w, δ), (vo, γο)〉. (a) If γ 0 < 0, then we may assume that γ 0 = − 1. Therefore, for each i ∈ I with k,¯¹(β) ∈ R, ⟨ u, v o ⟩ − α > 0 ≥ ⟨ w i, v o ⟩ − k,¯¹(β), and hence, ⟨ u, v o ⟩ > α and for each i ∈ I , k − 1 (β) ≥ ⟨ w i , v 0 ⟩ . Then, k i ⟨ ⟨ w i , v 0 ⟩ ≥ β , that is, f (v 0 ) ≤ β , which contradicts (i). (b) If γ 0 = 0, then for each i ∈ I with k i − 1 (β) ∈ R, { u, v 0 } > 0 ≥ { w i, v 0 } , and since L(f, ≤, β)̸= ∅, there exists x 0 ∈ L(f, ≤, β). Then, for all t > ∅, for each i ∈ I with k i − 1 (β) ∈ R, 〈 w i, x 0 + tv 0 〉 ≤ 〈 w i, x 0 〉 , which implies k i 〈 〈 w i, x 0 + tv 0 〉 ≤ k;({wi,xo}) ≤ β . Therefore, xo + tvo ∈ L(f, ≤,β). However, since { u,vo} > 0, there exists t 0 > 0 such that 〈 u, x 0 + t 0 v 0 〉 > α . This is contradiction. (c) If γ 0 > 0, we can assume that γ 0 = 1, then for each i ∈ I with k i − 1 (β) ∈ R and δ ≥ k − 1 (β), { u, v 0 } + α > 0 ≥ { w i, v 0 } + δ . However, this is a contradiction for ■ sufficiently large δ. A generator of the quasiconvex function obtained in Theorem 3.27 is not unique, and any lsc quasiconvex function has infinite generators. However, the set in condi- tion (ii) of Theorem 3.28 does not depend on the generator of the function f .  When {(k i , w i ) | i ∈ I} ⊂ Q × X* and {(l j , u j ) | j ∈ J} ⊂ Q × X* are generators of f , the following three conditions can be proven to be equivalent by using Theorem 3.28:
Moreover, the set in the above condition (ii) is equal to epiδ å , that is
where A = L( f, ≤, β ).  This equation is very important to define the newly proposed CCCQ in the way described in [7]. Next, we present a characterization of the containment of a convex set, defined by a quasiconvex constraint, in a reverse convex set, as defined by a convex constraint. Theorem 3.29. Let f be a lsc quasiconvex function from X to R with generator {(ki, wi) | i ∈ I} ⊂ Q × X*, and let h be a proper lsc convex function from X to R ∪ {+∞}.  In addition, let β ∈ R.  Assume that A = {x ∈ X | f(x) ≤ β} ≠ ∅, domh ∩ A̸= ∅, epih* + epiδ* is w*-closed and that there exists i 0 ∈ I such that k; 1 ( β ) ∈ R. Then, (i) and (ii) given below are equivalent:
Proof. Assume the condition (ii) holds. Based on the above assumption, subdiffer- ential sum rule, and Theorem 3.28, we can show that
This shows inf x ∈ A h ( x ) ≥ 0, that is, condition (i) holds. The inverse implication is similar. 3.5. Discussion. We compare the main results in this section with previous ones in Refs. [6, 8, 10, 14]. Consider the sets
and
where I, J, E, S and W are arbitrary sets, I ∪ J ∪ E̸= ∅, S ∪ W̸= ∅, and { f i,i ∈ I }, { g j,j ∈ J }, { l e,e ∈ E }, { k s,s ∈ S } and { h w,w ∈ W } are functions from X to R. We summarize in Table 1 the results on set containments in the similar way in [8]. No. from 1 to 7 are previous results, No. 8 to 13 are our results in this section. The column 2 inform about reference or section. The columns 3, 4, 5, 6 and 7 inform on the cardinality of the index sets, which can be empty, finite or arbitrary (abbreviated as " f " , "Fin" and " Arb", respectively), and the columns 8, 9, 10, 11 and 12 inform about assumptions of functions, which can be affine, quadratic concave, differentiable convex, differentiable concave, convex, concave, quasiconvex and quasiconcave (abbreviated as "Aff", "Quad", "Dconv", "Dconc", "Conv", "Conc", "Qconv" and "Qconc", respectively). "***" means that J U W  ≠ 0.
In the rest of the section, we compare No. 4 with No. 8 especially. Section 3.1 Table 1. Literature on set containments characterizes the containment in the form L(f,<,β) ⊂ {x | {v,x} < α}, whereas, in Theorem 3.1, Jeyakumar considered inclusions of the form L(f, <, β) ⊂ { x | { v, x } ≤ α }. We discuss conditions guaranteeing the equivalence of both inclusions. It is easy to show that for any v ∈ R n \ {0}, int{x | {v,x} < α} = {x | {v,x} < α}, Moreover, if f is continuous, we have cl{x | ⟨v,x⟩ < α} = {x | ⟨v,x⟩ ≤ α}. L(f, <, β) ⊂ intL(f, ≤, β),   clL(f, <, β) ⊂ L(f, ≤, β) are satisfied, but the converse inclusions are not true in general. When the equalities are fulfilled in these inclusions, we can show easily that our form and Jeyakumar's form are equivalent. For our purpose, we show the following lemmas: Lemma 3.30. Let A, B ⊂ R n . If intA = ∅ and int(clB) = intB, then we have int( A ∪ B ) = int B . Proof. Inclusion int( AU B ) ⊃ int B is obvious. Conversely, for any x ∈ int( AU B ), there exists r > 0 satisfying B(x,r) ⊂ A ∪ B. If int(B(x,r) ∩ B c )̸= ∅, then we have a contradiction since int A = ∅ and B ( x, r ) ∩ B c ⊂ ( A ∪ B ) ∩ B c ⊂ A hold. Therefore ∅ = int( B ( x, r ) ∩ B c ) = B ( x, r ) ∩ int( B c ) = B ( x, r ) ∩ (clB) c, and then B ( x, r ) ⊂ clB . By using assumption int(clB ) = int B , we obtain B ( x, r ) ⊂ int(clB) = int B ⊂ B . This shows that x ∈ int B . ■ Lemma 3.31. Let A, B ⊂ R n . If intA = ∅ and B is a convex set with intB̸= ∅, then we have (i) int( A ∪ B ) = int B , (ii)  int( A ∪ B c ) = int B c . Proof. Since B is convex and int B ≠ 0, we have int(clB) = intB and cl(intB) = clB. The second equation yields int(cl( B c )) = int( B c ). Therefore (i) and (ii) are proved by using Lemma 3.30. Theorem 3.32. Let f be a continuous quasiconvex function from R n to R, v ∈ R" \ {0} and α ∈ R.  If intL(f, =,β) = θ and intL(f, <,β) ≠ θ for some β ∈ R, then we have (i) L(f,<,β) = intL(f,<,β), (ii) clL(f,<,β) = L(f,<,β). Moreover Proof. (i). Put A = L(f,=,β) and B = L(f,<,β). By using Lemma 3.31 (i), intL(f,<,β) = intL(f,<,β) = L(f,<,β) because f is usc. Next we show (ii). Put A = L(f,=,β) and B = L(f,<,β). By using the Lemma 3.31 (ii), we have intL(f, ≥, β) = intL(f, >, β), and equivalently clL(f, <, β) = clL(f, ≤, β). Since f is lsc, clL( f, ≤, β) = L( f, ≤, β). The equivalence is straightforward consequence of statements (i) and (ii). ■ Remark 3.33. If every f i is convex, dom(sup i ∈ I f i ) = R n and condition [A1] in Theorem 2.8 holds, then the assumptions of Theorem 3.4 is satisfied.  Also if inf  sup f i (x) < 0, we can check that intL (sup i ∈ I f i ; =, 0) = ∅ and intL (sup i ∈ I f i ; <
x∈ R n i∈ I For any α ∈ (0, ∞) and any ν ∈ R n \ {0}, we have the following ,0) ≠ 0. characterization concerned with the Fenchel conjugate and H -quasiconjugate:
4. C ONSTRAINT QUALIFICATIONS We consider the following mathematical programming problem: minimize f ( x ), | subject to gi(x) ≤ 0,Vi ∈ I, where I is an arbitrary set, f and g i are extended real-valued functions from locally convex Hausdorff topological vector space X. In convex programming, a constraint qualification is an essential ingredient of the elegant and powerful duality theory. The best-known constraint qualifications are the Slater-type constraint qualifications. Often, however, such constraint quali- fications are not satisfied for problems that arise in applications. The lack of a con- straint qualification can cause theoretical and numerical difficulties in applications. For convex programming, Jeyakumar, Dinh, and Lee [11] developed the closed cone constraint qualification involving epigraphs and extending the Slater-type condi- tions. Constraint qualifications involving epigraphs have been used extensively in various studies, see [1, 7, 13]. Such constraint qualifications concern Jeyakumar's set containment characterization (Theorem 3.1). Recall Γ n (X), the set of all proper lsc convex functions from X to R. In [7], the condition of Farkas-Minkowski (FM) was investigated as the weakest constraint qualification for Lagrange (strong) duality. Let I be an index set, for each i ∈ I, let g i ∈ Γ 0 ( X ). The convex system { g i ( x ) ≤ 0 | i ∈ I } is said to be FM if the characteristic cone
is w*-closed. The following theorem indicates that FM is the weakest constraint qualification for Lagrange duality. Theorem 4.1. [7] Let I be an index set, for each i ∈ I, let g i ∈ Γ 0 (X). Assume that A = {x ∈ X | Vi ∈ I, gi (x) ≤ 0}̸≠ ∅. Then, the following statements are equivalent:
where R{1) = {λ ∈ R 1 | Vi ∈ I,λ i ≥ 0,{i ∈ I | λ i ≠ 0} is finite}. FM and Theorem 4.1 is closely related to Jeyakumar's set containment charac- terization (Theorem 3.1). In [13], Li, Ng and Pong investigated the basic constraint qualification for convex programming. Let { g i | i ∈ I } ⊂ Γ 0 ( X ), then the family { g i | i ∈ I } is said to satisfy the basic constraint qualification (the BCQ) at x ∈ A if
where A = { x ∈ X | Vi ∈ I, g i ( x ) ≤ 0} and I ( x ) = { i ∈ I | g i ( x ) = 0}. Also, they investigated the following theorem which indicates that the BCQ is the weakest constraint qualification for a certain optimality condition. Theorem 4.2. [13] { g i | i ∈ I } ⊂ Γ o ( X ). Assume that A = { x ∈ X | Vi ∈ I, gi(x) ≤ 0}̸= ∅ and x 0 ∈ A. Then, the following statements are equivalent: (i) {gi(x) ≤ 0 | i ∈ I} satisfies the BCQ at x 0 ,
(iii) for all f ∈ Γ 0 (X) with domf ∩ A̸= ∅ and epif* + epiδ* is w*-closed, x 0 is a minimizer of f in A if and only if there exists λ ∈ R (1 (xo)) such that
In this section, we investigate two constraint qualifications for a Lagrange-type These duality and a newly optimality condition on quasiconvex programming. constraint qualifications is similar to FM and the BCQ. This section is based on [24, 23]. 4.1. Closed cone constraint qualification. In this section, we show a duality theorem for quasiconvex programming with the new constraint qualification which deals with the set containment characterization by the generator (Theorem 3.28). At first, we introduce a new closed cone constraint qualification for quasiconvex programming. Definition 4.3. Let g be a lsc quasiconvex function from X to R with a generator {(ki, wi) | i ∈ I} ⊂ Q × X*, A = {x ∈ X | g(x) ≤ 0}̸≠ ∅ and there exists i 0 ∈ I such that k i − 1 (0) ∈ R. Then, the quasiconvex system { g ( x ) ≤ 0} satisfies the closed cone constraint qualification for quasiconvex programming (the Q-CCCQ) w.r.t. {(k i , w i | i ∈ I } if
is w*-closed. As a consequence of Theorem 3.28, {g(x) ≤ 0} satisfies the Q-CCCQ w.r.t. {(k i , w i ) | i ∈ I } if and only if the alternative form of the Q-CCCQ,
holds. In the remainder of the present paper, we fix a generator {( k i , w i ) | i ∈ I} ⊂ Q × X* of g and assume that there exists i 0 ∈ I such that wio = 0 and
This assumption is trivial because {( k i , w i ) | i ∈ I \ { i 0 }} is also generator of f . However, the assumption is important because it ensures the following inclusion, which is critical in the proof of Theorem 4.4:
Theorem 4.4. Let g be a lsc quasiconvex function from X to R with a generator {(k i , w i ) | i ∈ I} ⊂ Q × X*.  Assume that A = {x ∈ X | g(x) ≤ 0} ≠ ∅.  Then, the following statements are equivalent: (i) {g(x) ≤ 0} satisfies the Q-CCCQ w.r.t. {(ki, wi) | i ∈ I}, (ii) for all v ∈ X*,
Proof. First, we prove that (i) implies (iii). Let f be a lsc convex function with inf x∈A f(x) ∈ R , where domf ∩ A̸= ∅ and epif* + epiδ*, is w*-closed. Then, as a result of Fenchel duality,
Therefore, there exists v 0 ∈ X* such that inf x ∈ A f (x) = − f * (v 0 ) − δ λ (− v 0 ). Fur- thermore, there exist λ ∈ R ( γ and δ ∈ R ( γ ) such that −ν 0 = Σ i ∈ 1 λ i w i , for each i ∈ I with λ i̸= 0, k − 1 (0) ≤ δ i , for each i ∈ I with λ i = 0, δ i = 0, and δ i (− v 0) = Σ i ∈ I λ i δ i . If there exists i 0 such that λ i 0̸= 0 and δ i 0 > k i 0 1 (0), then there exists γ ∈ R such that γ < δ * ( − ν 0 ) and ( − ν 0 , γ ) ∈ epiδ * , which is a contradiction. Therefore, for each i ∈ I with λ i̸= k i − 1 (0) and (iii) holds. Next, we show that (iii) implies (ii). Let v ∈ X* with inf z ∈ A 〈 V, x 〉 ∈ R. Then, because of condition (iii), there exists λ ∈ R ( 1 ) such that
where h is the Fenchel conjugate of v . Clearly, v = − Σ i ∈ I λ i w i . Therefore, (ii) holds. Finally, we show that (ii) implies (i). Let {(vk,αk)} ⊂ coneco ∪ i ∈ I {(wi,δ) ∈ X* × R  | k − 1 (0) ≤ δ} with ( vk ,αk ) converging to ( v 0 ,α 0 ).  Since ( v 0 ,α 0 ) ∈ cl cone co U i ∈ 1  {(w i, δ)  ∈ X* × R  | k i − 1 (0)  ≤  δ}, by Theorem 3.28, we have (vo, αρ) ∈ epiδ*, Hence,
If v 0 = 0, then we can prove that α 0 ≥ 0. From inclusion (1),
If v 0̸= 0, then by (ii), we show that there exists λ ∈ R ( η such that v 0 = Σ i ∈ 1 λ i w i and inf z ∈ A − V 0 (x) = − ∑ i ∈ I λ i k − 1 (0). Therefore ∑ i ∈ I λ i k − 1 (0) ≤ α 0 . Because v 0̸≠ 0, there exists i 0 ∈ I such that λ i0 > 0. Then, we set δ ′ ∈ R ( l ) as follows. For each i ∈ I with λ i̸= 0 and i̸= i 0 , δ i = k i − 1 (0), and for each i ∈ I with λ i = 0, δ i = 0 and δ i o = k i o 1 (0) + ∞− ει∈ ιλ i k − 1 (0)  Then α o = Σ i ∈ ιλ i δ i , that is,
Next, we explain the usefulness of the Q-CCCQ. We give the following important example. Example 4.5. Let X = R n , g : R n → R , g(x) = ξx 2 , and consider the system g* : Rn → R, g*(v) = ½v², and cone co epig* = ( R n × (0, ∞ )) ∪ { x ∈ R n | x 1 = 0}, that is, {g(x) ≤ 0} is not FM. However, we can choose a generator which satisfies the Q-CCCQ. Let B = {(1,0, . . . ,0), (−1,0, . . . ,0)} ⊂ R n , k ∈ Q as follows:
Then, g = sup w ∈ b k ⊙ w, k − 1 (0) = 0, and coneco Uweb {(w, δ) | k-1(0) ≤ δ} = R n × [0, ∞) hold, that is, {g(x) ≤ 0} satisfies the Q-CCCQ w.r.t. {(k,w) | w ∈ B}. Also, {⟨ w,x ⟩ − k − 1 (0) ≤ 0 | w ∈ B } is FM, hence we can use Theorem 4.1 and Theo- rem 4.4 to this example. Finally, we give the following example for quasiconvex but not convex problem. Example 4.6. Let X = R n , a ∈ R n , g : R n → R , g(x) = √ ∥ x − α ∥ − 2, and
consider the system { g(x) ≤ 0}. Then, we can check that A = { x ∈ R n | ∥ x − α ∥ ≤ 4}. Let B = { w ∈ R n | ||w|| = 1}, kw ∈ Q as follows:
, Then, g = sup w ∈ B k w ◦ w , k − 1 (0) = ⟨ w , a ⟩ , and epiδ λ = coneco ∪ w ∈ B {(w , δ ) | k = 1 (0) ≤ δ} hold, that is, { g ( x ) ≤ 0} satisfies the Q-CCCQ w.r.t.  {( kw, w ) | w ∈ B }. Hence, we can use Theorem 4.4 to this example. Of course, { 〈 w, x 〉 − k − 1 (0) ≤ 0 | w ∈ B } is also FM, hence we can use Theorem 4.1 to this examples. In any case, it is very profitable and useful to be able to represent constraint functions by using the notion of generator. 4.2. Optimality conditions and the basic constraint qualification.  The pur- pose of this section is to generalize Theorem 4.2 for quasiconvex programming.  By the notion of generator, we introduce a new subdifferential for quasiconvex func- tions, and by using this subdifferential, we investigate generalized results reported in previous studies. At first, we introduce the new subdifferential for quasiconvex functions. Definition 4.7. Let f be a lsc quasiconvex function with a generator G = {(ks,ws) | s ∈ S} ⊂ Q × X*, and assume that ks is lower left-hand Dini dif- ferentiable for each s ∈ S . Then, we define the subdifferential of f at x 0 with respect to G as follows: where S(xo) = {s ∈ S | f(xo) = k s o ws(xo)}. This subdifferential is a generalized notion of the subdifferential for convex func-
Also, if f is Gâteaux differentiable at x 0 , k s are differentiable at 〈 w s , x 0 〉 for each s ∈ S(x 0), and S(x 0)̸= ∅, then we can check θ G f(x 0) = { f ′ (x 0)}. Actually, for each s ∈ S(x 0) and d ∈ X,
Similarly, we can prove that 〈f ′ ( x 0 ), − d ⟩ ≥ 〈 k ′ (〈 w s, x 0 〉 w s, − d 〉 , that is, f ′ ( x 0 ) ＝ k',({ws, x0})ws. Next, we show a necessary condition for a minimizer of a certain quasiconvex function in a closed convex set. Theorem 4.8. Let A be a closed convex subset of X , f be a lsc quasiconvex function with a generator G = {(ks,ws) | s ∈ S} ⊂ Q × X*. Assume that ks is lower left- hand Dini differentiable for each s ∈ S and at least one of the following holds: (i) S is finite and k s is continuous for each s ∈ S , (ii) X is a Banach space, S is a compact topological space, s → w s is continu- ous on S to (X*, || · ||), (s,t) ↔ k s (t) is usc on S × R, and (s,t) ↔ D_k s (t) is continuous on S × R. If x 0 is a local minimizer of f in A then, Proof. At first, we show that ∂ G f ( x 0 ) is w*-compact. It is clear when the condition (i) holds. If the condition (ii) holds, then S ( x 0 ) is compact because S ( x 0 ) = { s ∈ S | f(x 0) ≤ k , ০ w s(x 0)} and s ↔ k , ০ w s(x 0) is usc on S . Thus, { D_k s({w s, x 0}}w s { s ∈ S(x 0)} is bounded since s ↔ w s is continuous on S and (s,t) ↔ D_k s (t) is continuous on S × R.  Hence, ∂ςf ( x 0 ) is w*-compact by the Banach-Alaoglu theorem. Now we assume that 0 f ∂Gf(xo)+ N A (xo). Since ∂Gf(xo)+ N A (xo) is w*-closed, we can find d 0 ∈ X \ {0} satisfying
for all y* ∈ ∂ G f (x 0) and x* ∈ N A (x 0). If s ∈ S (x 0), then D − k s ( { w s, x 0}) > 0 and ⟨ w s, d 0 ⟩ < 0 since D_k s ( { w s, x 0 } )w s ∈ ∂ c f (x 0 ) and k s is non-decreasing. From this, we have sup s∈S(x 0 ) { W s, d 0 } < 0 and d } → sup s∈S(x 0 ) { W s, d } is usc. Indeed, it is clear when the condition (i) holds. If the condition (ii) holds, we can check them since S ( x 0 ) is compact and s → w s is continuous on S . Therefore, there exists U do a neighborhood of do such that { w s, d } < 0 for each s ∈ S ( x 0 ) and d ∈ U d 0 . Since x 0 is a local minimizer of f in A , there exists U z 0 a neighborhood of x 0 such that for all x ∈ U x 0 ∩ A, f(x 0 ) ≤ f(x). Also d 0 ∈ T A ( x 0 ) = cl ∪ λ > 0 ∧ z z because ⟨ x * , d 0 ⟩ ≥ 0 for all x * ∈ N A ( x 0 ).
Then there exist d 1 ∈ U d 0 , λ 0 > 0 and x 1 ∈ A such that d 1 = ≥ x = − − − (1 − π ) x 0 + π x 1 = x 0 + π d 1 , then x n ∈ A ∩ U x 0 for large enough n , therefore f(xo) ≤ f(xn). If the condition (i) holds, since S is finite, we can find s 0 ∈ S and a subsequence { x n,} of { x n } such that s 0 ∈ S ( x n, ) for each i ∈ N, and we have s 0 ∈ S ( x 0 ) because f and k so 0 w so are continuous. For large enough i ∈ N, k so 0 w so (x n i ) = f (x n i ) ≥ f(xo) = k s 0 w s 0 (xo), and then,
since d 1 ∈ U d 0 . Therefore D_k s 0 ( \ w s 0 , x 0 ) ≤ 0, it is contradiction. If the condition (ii) holds, all S ( x n ) are not empty because S is compact and s → k s o w s ( x n ) is usc on S . Let { s n } be a sequence satisfying s n ∈ S ( x n ) for all n ∈ N. Then there exists a subsequence { s n i } of { s n } such that { s n i } converges to some so ∈ S . Therefore
that is, s 0 ∈ S ( x 0 ). Then, for sufficiently large i ∈ N, k sni ◦ W sni ( xni ) = f ( xni ) ≥ f(x 0) ≥ k sn i o w sn i (x 0) and ( w sn i ,d 1 ) < 0, because s 0 ∈ S (x 0),d 1 ∈ U d 0 and { w sn i } converges w so . From this and k sn , is non-decreasing, k sn , is constant on interval [〈Wsnç, xo〉+ π i 〈Wsnç, dı〉,〈Wsnç, xo〉] and hence we have D_ksnç(〈Wsnç, xo〉) = 0. ■ Finally we obtain D_kso({wso, xo}) = 0, but this is a contradiction. On the other hand, in separable Banach space, a similar result was introduced when S is compact, f s are locally Lipschitz, f = sup s ∈ S f s , and certain assumptions hold in [18]. If condition (ii) holds and k s are differentiable, then k s 0 w s are locally Lipschitz. However, in Theorem 4.8, we assume that X is a usual Banach space and k s are only lower left-hand Dini differentiable, thus, Theorem 4.8 is not a direct consequence of the result in [18]. Also, if f is a proper lsc convex function with basic generator B f and dom f* is compact, then condition (ii) holds. For this reason, it seems that condition (ii) is not so strong for quasiconvex programming. We define a new constraint qualification and consider an optimality condition for quasiconvex programming with inequality constraints, and we prove that the new constraint qualification is the weakest constraint qualification for the optimality condition. At first, we introduce the following new constraint qualification. Definition 4.9. Let {g; | i ∈ I} be a family of lsc quasiconvex functions from X to R, for each i ∈ I, {(k( i,j),W(i,j) | j ∈ J,} ⊂ Q × X* be a generator of g i, T = {t = (i,j) | i ∈ I,j ∈ Ji}, T(x) = {t ∈ T | ki((wt,x)) = 0, ki − 1 (0) = { wt,x)}, and A = {x ∈ X | ∀i ∈ I, gi(x) ≤ 0}. The family { g i | i ∈ I } is said to satisfy the basic constraint qualification for quasiconvex programming (the Q-BCQ) with respect to {( k t , w t ) | t ∈ T } at x ∈ A if
We can check that one inclusion always holds. Indeed, for each t ∈ T ( x ) and y ∈ A, ⟨ w t,y ⟩ ≤ ⟨ w t,x ⟩ because ⟨ w t,x ⟩ = k − 1 (0). Furthermore, N A ( x ) is a convex cone, this shows that N A ( x )つconeco U t ∈ T ( x ) { W t } .  Therefore, the Q-BCQ is equivalent to the following inclusion
In the following theorem, we show an optimality condition for quasiconvex pro- gramming and the Q-BCQ is the weakest constraint qualification for this optimality condition. Let Q F (X) be the set of all quasiconvex functions which have a finite, continuous and lower left-hand Dini differentiable generator, that is,
Theorem 4.10. Let { g i | i ∈ I} be a family of lsc quasiconvex functions from X to R, for each i ∈ I, {(k( ij ), W(ij ) | j ∈ J i } ⊂ Q × X* be a generator of g i , T = {t = (i,j) | i ∈ I,j ∈ J i }, T(x) = {t ∈ T | k i ((w t,x)) = 0, k − 1 (0) = { w t,x)}, A = { x ∈ X | ∀i ∈ I, gi ( x ) ≤ 0} and x 0 ∈ A. Then, the following statements (i), (ii), (iii) and (iv) are equivalent: (i) {g i (x) ≤ 0 | i ∈ I} satisfies the Q-BCQ w.r.t. {(k t,w t ) | t ∈ T} at x 0, (ii) for each ν ∈ X*, x 0 is a minimizer of ν in A if and only if there exists λ ∈ R ( T ) such that λ t = 0 for each t ∈ T \ T ( x 0 ), the complementarity condition, and
(iii) for each f ∈ Γ 0 (X) with domf ∩ A̸= ∅ and epif* + epiδ* is w*-closed, x 0 is a minimizer of f in A if and only if there exists λ ∈ R ( T ) such that λt = 0 for each t ∈ T \ T(x 0), and
(iv) for all f ∈ Q F (X) with a generator G, if x 0 is a local minimizer of f in A, then, there exists λ ∈ R ( T ) such that λ t = 0 for each t ∈ T \ T ( x 0), and
Proof. We now first prove (i) implies (iii). By the assumption of f , the subdiffer- ential sum formula holds, that is, ∂(ƒ + δ A )(x 0) = ∂ƒ(x 0) + ∂ A (x 0) Because ∂δ A (x 0 ) = N A (x 0 ) and condition (i) holds, x 0  minimizes  f  on  A  ⇐⇒  0 ∈ ∂f (x 0 ) + cone co
this shows that (iii) holds. Next, it is clear that (iii) implies (ii) and (iv) implies (ii). We now prove that (ii) implies (i). We want to show that if x* ∈ N A (x 0 ) then x* ∈ coneco U t ∈ T ( x 0 ) { W t } .  Let x* ∈ N A ( x 0 ) .  Because x* ∈ N A ( x 0 ), δ λ ( x* )  = 〈 x *, x 0〉. Therefore, x 0 minimizes − x * on A . Then by using condition (ii), there exists λ ∈ R ( T ) such that x * = ∑ t ∈ T λ t W t ∈ cone co ∪ t ∈ T ( x 0 ) { W t } Finally, by using Theorem 4.8, we can prove (i) implies (iv). This completes the ■ proof. In Theorem 4.10, Q F (X) corresponds to the condition (i) of Theorem 4.8. In the following theorem, we define Qc(X) which corresponds to the condition (ii) of Theorem 4.8 as follows,
Theorem 4.11. Let { gi | i ∈ I} be a family of lsc quasiconvex functions from X to R, for each i ∈ I, {(k( ij ), W(ij ) | j ∈ J i } ⊂ Q × X* be a generator of g i, T = {t = (i,j) | i ∈ I,j ∈ Ji}, T(x) = {t ∈ T | ki((wi,x)) = 0, ki 1 (0) = { wi,x)}, A = { x ∈ X | ∀ i ∈ I, g i ( x ) ≤ 0} and x 0 ∈ A. Assume that X is a Banach space, then, the following statements (v) is equivalent to the statements (i), (ii), (iii) and (iv) in Theorem 4.10. (v) for all f ∈ Qc(X) with a generator G = {(ks,ws) | s ∈ S} ⊂ Q × X*, if x 0 is a local minimizer of f in A, then, there exists λ ∈ R ( T ) such that λ t = 0 for each t ∈ T \ T(x 0), and
Proof. By using Theorem 4.8, we can prove (i) implies (v). Also, it is clear that ■ (v) implies (ii). Next, we investigate a relation between the Q-BCQ and the BCQ. Let { g i | i ∈ I } be a family of proper lsc convex function with the basic generator, T = {(i,v) | i ∈ I , v ∈ domg ∗ }, then, for all x ∈ A , we can check
that is, the BCQ and the Q-BCQ w.r.t. the basic generator are equivalent. Fur- thermore, we can prove Theorem 4.2, by using Theorem 4.10. Also, we can prove that the conditions (i), (ii) and (iii) in Theorem 4.10 are equivalent by using Theorem 4.2. Let { g i | i ∈ I } be a family of lsc quasiconvex functions from X to R, for each i ∈ I, {(k(i,j),w(i,j)} | j ∈ Ji} ⊂ Q × X* be a generator of g i , and T = {t = (i,j) | i ∈ I,j ∈ J i } . Then, A = {x ∈ X | ∀t ∈ T, wt(x) − k − 1 (0) ≤ 0} and ∂(wt − k − 1 (0)) = { wt} for each t ∈ T. Since w t − k − 1 (0) is a continuous linear function, we can prove an equivalence relation of the conditions (i), (ii) and (iii) in Theorem 4.10 by using Theorem 4.2. Hence, we can see that (i), (ii) and (iii) of Theorem 4.10 and Theorem 4.2 are equivalent. However, (iv) of Theorem 4.10 and (v) of Theorem 4.11 are new results which concern quasiconvex programming and we can consider problems whose objective function is quasiconvex by using Theorem 4.10 and 4.11. In the last of this section, we emphasize the usefulness of optimality conditions and Q-BCQ by some examples. At first, we show the following quasiconvex pro- gramming problem that Theorem 4.10 is used effectively. Example 4.12. Let X = R 2 , I = {1,2}, g1(x) = −(x1 − 2) 3 , g2(x) = −(x2 − 1) 5 and f(x) = √ [ x 1 − 1] + [ x 2 − 1], then, f , g 1 and g 2 are continuous quasiconvex, and A = { x ∈ R 2 | x 1 ≥ 2, x 2 ≥ 1}.  Also, G 1 = {( k 1 , (−1,0)) | k (a) = (a + 2) 3} is a generator of g 1 , G 2 = {(k 2 , (0 , −1)) | k 2 (a) = (a + 1) 5 } is a generator of g 2 and G 0 = {(h1, (1, 1)), (h2, (−1, 1)), (h3, (−1, −1)), (h4, (1, −1))} is a generator of f , where h 1 be a function from R to R as follows:
and h 2 (a) = h 4 (a) = h 1 (a + 2), h 3 (a) = h 1 (a + 4) for all a ∈ R .  We can check easily that the Q-BCQ w.r.t. G 1 ∪ G 2 is satisfied at each point of A . We observe whether there exist x ∈ A and λ ∈ R 2 satisfying 0 ∈ ∂ G 0 f (x) + λ 1 (−1,0) + λ 2 (0, −1) and the complementarity condition or not. If x ∈ int A , then, ∂ Gof (x) = { 2 √ 21 + x 2 − z (1, 1)} and g i ( x )̸= 0 ( i ∈ I ), this implies λ = 0 if the complementarity condition holds.  Hence, the optimality condition is not satisfied.  If x ∈ {y | y1 = 2,y2 > 1}, then, λ2 = 0 if the complementarity condition holds.  Also, ∂Gof(x) = { z √ x 1 + z = z (1,1)}, that is, the optimality condition is not satisfied. If x ∈ {y | y1 > 2,y2 = 1}, then, ∂ς o f (x) = cl co { 2√x1+z2 = z (1,1), 2√x1 = z (1, − 1) } and λ 1 = 0 if the complementarity condition holds, that is, the optimality condition is not satisfied. If x = (2, 1), then,
the necessary condition for a local minimizer. In this case, the other x ∈ A does not satisfy the optimality condition, hence (2, 1) is the global minimizer of f in A . As stated above, Q-BCQ is used effectively for quasiconvex programming.  At the same time, Q-BCQ is useful for convex programming. Now we show the following example that the Q-BCQ is satisfied and the BCQ is not satisfied. This example indicates that the range of functions which satisfies some of constraint qualifica- tions is extended. This extension is very important because the lack of constraint qualifications can cause theoretical and numerical difficulties in applications. Example 4.13. Let X = R 2 , I = {1}, g(x) = (x 1 − x 2 ) 2 . Then, A = {y | y 1 = y 2 }, for all y ∈ A, N A (y) = { v | v 1 + v 2 = 0}, I (y) = I. Also, the BCQ is not satisfied at any point y ∈ A because ∇ g ( y ) = 0. However, we can choose a suitable generator for satisfying the Q-BCQ. Let k be a function from R to R as follows:
let J = {(k, (1, −1)), (k, (−1,1))}. Then, J is a generator of g. Furthermore, for all y ∈ A, k(\(1, −1), (y1,y2)}) = k(\(−1,1), (y1,y2)}) = 0, and N A (y) = { v | v 1 + v 2 = 0} = cone co ∪{(1, −1), (−1, 1)}. Therefore the Q-BCQ w.r.t. J at y is satisfied. Let f(x) = (x1 − 5)² + (x2 − 3)², then, f is a continuous convex function. Since Q-BCQ is satisfied, we can find a minimizer by using an optimality condition in this paper.  We observe whether there exist x ∈ A and λ ∈ R 2 satisfying 0 ∈ ∂f ( x ) + λ 1 (1, −1) + λ 2 (−1, 1) and the complementarity condition or not. We can check easily that ∂f (x) = {∇f (x)} = {(2(x1 − 5), 2(x2 − 3))}.  If there exists λ satisfying the optimality condition, then, we can calculate x = (4,4). Put λ = (0,2), then 0 ∈ ∂f ( x ) + λ 1 (1, −1) + λ 2 (−1,1).  By using Theorem 4.10, (4,4) is the global minimizer.  Also, let g 1 = 〈(1, −1), x 〉 and g 2 = 〈(−1,1), x 〉 , then A = { x ∈ R 2 | g i ( x ) ≤ 0( i = 1, 2)} and { g i ( x ) ≤ 0 | i = 1, 2} satisfies BCQ. Hence, we can use Theorem 4.2 to this example. Acknowledgment. The author is grateful to a referee for many comments and sug- gestions improved the quality of the paper.