Solving Quadratic Integer Programs: Small Changes Yield Big Solving Quadratic Integer Programs: Small Improvements Changes Yield Big Improvements Yong Xia Outline Introduction Yong Xia Quadratic Convex Reformulation Beihang University Probabilistically Constrained Quadratic dearyxia@gmail.com Programs Box- Sep. 2, 2014 Constrained Nonconvex Quadratic Integer Program Thanks
Outline Solving Quadratic Integer Programs: Small Changes Yield Big 1 Introduction Improvements Yong Xia Outline 2 Quadratic Convex Reformulation Introduction Quadratic Convex 3 Probabilistically Constrained Quadratic Programs Reformulation Probabilistically Constrained Quadratic Programs 4 Box-Constrained Nonconvex Quadratic Integer Program Box- Constrained Nonconvex Quadratic Integer Program Thanks
Quadratic Constrained Quadratic Programming (QCQP) Solving Quadratic Integer Programs: Small Changes (QCQP): Yield Big Improvements min f ( x ) := x T Ax + 2 a T x Yong Xia (1a) s . t . h i ( x ) := x T B i x + 2 b T i x + c i = ( ≤ )0 , i = 1 , . . . , m . (1b) Outline Introduction Quadratic Convex Special cases: Binary Quadratic Program as for binary Reformulation variables: Probabilistically Constrained ⇒ x 2 x i ∈ { 0 , 1 } ⇐ i − x i = 0 . Quadratic Programs NP-hard Box- Constrained Nonconvex Quadratic Integer Program Thanks
Lagrangian Dual Lagrange function: Solving Quadratic � Integer L ( x , µ ) = f ( x ) + µ i h i ( x ) Programs: Small Changes Yield Big i � � � Improvements x T ( A + µ i b i ) T x + = µ i B i ) x + 2( a + µ i c i , Yong Xia i i i Outline where µ i ≥ 0 for h i ( x ) ≤ 0. Introduction Lagrangian dual problem of (QCQP) has an explicit Quadratic formulation: Semidefinite programming (SDP): Convex Reformulation � � Probabilistically ( D ) sup inf x L ( x , µ ) Constrained µ Quadratic � Programs = sup µ i c i − s Box- Constrained i � A + � a + � � Nonconvex i µ i B i i µ i b i Quadratic a T + � � 0 , Integer s . t . i µ i b T s Program i Thanks where B � 0 stands for that B is positive semidefinite.
Strong Duality for m = 1& inequality: S-Lemma Solving Let f ( x ) = x T Ax + 2 a T x + c and h ( x ) = x T Bx + 2 b T x + d Quadratic Integer be two quadratics having symmetric matrices A and B . Programs: Small Changes Under the Slater assumption, i.e., there is an x ∈ R n such that Yield Big Improvements h ( x ) < 0, the quadratic system Yong Xia f ( x ) < 0 , h ( x ) ≤ 0 (2) Outline Introduction is unsolvable if and only if there is a nonnegative number µ ≥ 0 Quadratic Convex such that Reformulation f ( x ) + µ h ( x ) ≥ 0 , ∀ x ∈ R n . (3) Probabilistically Constrained Quadratic [1]Yakubovich, V.A.: S-procedure in nonlinear control theory. Programs Vestnik Leningrad. Univ. 1, 62 õ 77 (1971) (in Russian) Box- Constrained [2]Yakubovich, V.A.: S-procedure in nonlinear control theory. Nonconvex Quadratic Vestnik Leningrad. Univ. 4, 73 õ 93 (1977) (English Integer Program translation) Thanks
S-Lemma with equality Solving Suppose Slater condition holds for h ( x ) = 0, i.e., there are Quadratic Integer x ′ , x ′′ such that h ( x ′ ) < 0 < h ( x ′′ ). S-Lemma with equality Programs: Small Changes holds under one of the following additional assumptions: Yield Big Improvements (A) h ( x ) is strictly concave (or convex), i.e., B ≺ ( ≻ )0. Yong Xia (B) There is an η ∈ R such that A � η B . Outline (C) h ( x ) is homogeneous. Introduction [A]P´ olik, I., Terlaky, T. A survey of the S-lemma. SIAM Quadratic Convex Review, 49(3), 371-418 (2007) Reformulation Probabilistically [B]Beck, A., Eldar, Y.C.: Strong duality in nonconvex Constrained Quadratic quadratic optimization with two quadratic constraint. SIAM J. Programs OPTIM. 17(3), 844-860 (2006) Box- Constrained [C]Tuy, H., Tuan, H.D.: Generalized S-lemma and strong Nonconvex duality in nonconvex quadratic programming. J. Global Optim. Quadratic Integer 56(3):1045-1072 (2013) Program Thanks
S-lemma with equality: Our Result Solving Quadratic Integer Under the Slater Assumption that h ( x ) takes both positive and Programs: Small Changes negative values, the S-lemma with equality holds if h(x) is not Yield Big Improvements linear, i.e., B � = 0. Yong Xia (Note that S-lemma with equality for the case B = 0 is easy to Outline verify.) Introduction Quadratic Convex S-lemma with equality = ⇒ the classical S-lemma since B � = 0 Reformulation is satisfied when converting h ( x ) ≤ 0 into h ( x ) + t 2 = 0. Probabilistically Constrained Quadratic Programs [6] Y. Xia, S. Wang, R.L. Sheu, S-Lemma with Equality and Its Box- Applications, arXiv:1403.2816v2 (2014) Constrained Nonconvex http://arxiv.org/abs/1403.2816 Quadratic Integer Program Thanks
Generalized Trust-region subproblem Solving Quadratic Integer Programs: Small Changes min x T Ax + 2 a T x Yield Big (4a) Improvements s . t . α ≤ x T Bx ≤ β, (4b) Yong Xia Outline Introduction x T Ax + 2 a T x ( GTRS ) inf (5) Quadratic α ≤ x T Bx + 2 b T x ≤ β, Convex (6) s . t . Reformulation Probabilistically [7] R.J. Stern and H. Wolkowicz, Indefinite trust region Constrained Quadratic subproblems and nonsymmetric perturbations. SIAM J. Programs Optim., 5(2), 286–313 (1995) Box- Constrained [8]Pong, T.K., Wolkowicz, H.: The generalized trust region Nonconvex Quadratic subprobelm, Comput. Optim. Appl. 58, 273-322 (2014) Integer Program Thanks
Pong and Wolkowicz’s Result and Open Question Pong and Wolkowicz have shown strong duality holds for Solving Quadratic (GTRS) under the following assumption: Integer Programs: Small Changes Assumption Yield Big Improvements Yong Xia 1. B � = 0 . 2. (GTRS) is feasible. Outline Introduction 3. The following relative interior constraint qualification holds Quadratic Convex ( RICQ ) α < tr ( B � x + d < β, for some � X )+2 b T � x T . Reformulation X ≻ � x � Probabilistically Constrained 4. (GTRS) is bounded below. Quadratic Programs 5. The dual of (GTRS) is feasible. Box- Constrained Nonconvex Quadratic Under Assumptions 1,2,3, it is trivial to see Item 5 = ⇒ Item 4. Integer Program They have proved when b = 0, Item 4 = ⇒ Item 5. An open Thanks question was raised when b � = 0.
S-lemma with interval bounds Under the Slater Assumption that there exists an x ∈ R n such Solving Quadratic that α < h ( x ) < β , S-lemma with interval bounds holds when Integer Programs: B � = 0, i.e., the system f ( x ) < 0 , α ≤ h ( x ) ≤ β is unsolvable if Small Changes Yield Big and only if there is a number µ ∈ R such that Improvements Yong Xia f ( x ) + µ − ( h ( x ) − β ) + µ + ( α − h ( x )) ≥ 0 , ∀ x ∈ R n . Outline where µ + = max { µ, 0 } , µ − = − min { µ, 0 } . Introduction Corollary Quadratic Convex Reformulation Under Items 1 , 2 , 3 in Pong and Wolkowicz’s Assumption, Probabilistically strong duality holds for ( GTRS ) . Moreover, under Items 1 , 2 , 3 Constrained Quadratic in Pong and Wolkowicz’s Assumption, Items 4 and 5 are Programs equivalent. Box- Constrained Nonconvex [9]Shu Wang, Yong Xia, Strong Duality for Generalized Trust Quadratic Integer Region Subproblem: S-Lemma with Interval Bounds, 2014 Program Thanks working paper
Approximate Algorithms: an example Solving Quadratic Integer Rather than providing relaxations, SDP also has applications in Programs: Small Changes giving approximate algorithms. For example, Yield Big Improvements f ( x ) = x T Ax + 2 b T x ( ECQP ) min x ∈ R n (7) Yong Xia � F k x + g k � 2 2 ≤ 1 , k = 1 , . . . , m , (8) s . t . Outline Introduction where � g k � < 1 is assumed. Quadratic Convex The semidefinite programming relaxation of (ECQP) is Reformulation Probabilistically Constrained B • X ( SDP ) min Quadratic B k • X ≤ 0 , k = 1 , . . . , m , Programs s . t . Box- X n +1 , n +1 = 1 , X � 0 , X ∈ R ( n +1) × ( n +1) . Constrained Nonconvex Quadratic Integer Program Thanks
Approximate Algorithms: an example Solving Theorem (Tseng 2003) Quadratic Integer Programs: For (ECQP), we can generate a feasible solution in polynomial Small Changes Yield Big time satisfying Improvements Yong Xia (1 − γ ) 2 ( √ m + γ ) 2 · v ( SDP ) , f ( x ) ≤ (9) Outline Introduction where γ := max k =1 ,..., m � g k � . Quadratic Convex Reformulation Very recently, we can show that the m in (9) can be improved Probabilistically Constrained to �� √ 8 m + 17 − 3 � � Quadratic Programs min , n + 1 . Box- 2 Constrained Nonconvex [10]P. Tseng, Further results on approximating nonconvex Quadratic Integer quadratic optimization by semidefinite programming relaxation, Program Thanks SIAM Journal Optimization, 14, 2003, 268-283
Summary of Applications of SDP Solving Quadratic Integer Programs: Small Changes Yield Big Improvements Yong Xia Providing efficient relaxations Outline Strong duality for special QCQP Introduction Establishing approximate algorithms Quadratic Convex Providing high-quality reformulations (The remaining of Reformulation this talk) Probabilistically Constrained Quadratic Programs Box- Constrained Nonconvex Quadratic Integer Program Thanks
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