Quadratic programming problems arise frequently in various industrial application settings, inclu... more Quadratic programming problems arise frequently in various industrial application settings, including facility location, multiperiod refinery, and circle packing problems. The general nonconvex quadratically constrained quadratic programming (QCQP) problem is NP hard and presents a significant challenge. We present a tight linear relaxation scheme for QCQP that can be used in the context of a branch-and-bound global optimization algorithm for this problem. Most of the current approaches rely on convex relaxations of the QCQP problem which are either nonlinear [7, 2] or relaxing each nonconvex term separately [1, 5]. The main idea of our approach is to relax the entire quadratic constraints by linear underestimations. We propose a relaxation that is tighter than standard linear relaxation based on the convex envelope of multilinear functions [4, 3, 6] and develop an efficient mechanism to tighten the standard relaxation via a class of multilinear cutting planes. Computational experie...
... Dual bounds and optimality cuts for all-quadratic programs with convex constraints. ... Exact... more ... Dual bounds and optimality cuts for all-quadratic programs with convex constraints. ... Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations. ... linearly on the vector x i provided that all the remaining n−1 vector arguments are fixed. ...
At the intersection of nonlinear and combinatorial optimization, quadratic programming has attrac... more At the intersection of nonlinear and combinatorial optimization, quadratic programming has attracted significant interest over the past several decades. A variety of relaxations for quadratically constrained quadratic programming (QCQP) can be formulated as semidefinite programs (SDPs). The primary purpose of this paper is to present a systematic comparison of SDP relaxations for QCQP. Using theoretical analysis, it is shown that the recently developed doubly nonnegative relaxation is equivalent to the Shor relaxation, when the latter is enhanced with a partial first-order relaxation-linearization technique. These two relaxations are shown to theoretically dominate six other SDP relaxations. A computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations, while providing relaxation gaps averaging 3% as opposed to gaps of up to 19% for weaker relaxations, on 700 randomly generated problems with up to 60 variables. An SDP relaxation derived from Lagrangian relaxation, after the addition of redundant nonlinear constraints to the primal, achieves gaps averaging 13% in a few CPU seconds.
We consider global optimization of nonconvex problems containing multilinear functions. It is wel... more We consider global optimization of nonconvex problems containing multilinear functions. It is well known that the convex hull of a multilinear function over a box is polyhedral, and the facets of this polyhedron can be obtained by solving a linear optimization problem (LP). When used as cutting planes, these facets can significantly enhance the quality of conventional relaxations in general-purpose global solvers. However, in general, the size of this LP grows exponentially with the number of variables in the multilinear function. To cope with this growth, we propose a graph decomposition scheme that exploits the structure of a multilinear function to decompose it to lower-dimensional components, for which the aforementioned LP can be solved very efficiently by employing a customized simplex algorithm. We embed this cutting plane generation strategy at every node of the branch-and-reduce global solver BARON, and carry out an extensive computational study on quadratically constrained quadratic problems, multilinear problems, and polynomial optimization problems.
Increased use of graphene materials might ultimately lead to their release into the environment. ... more Increased use of graphene materials might ultimately lead to their release into the environment. However, only a few studies have investigated the impact of graphene-based materials on green plants. In this study, the impact of reduced graphene oxide (RGO) on the microalgae Scenedesmus obliquus was evaluated to determine its phytotoxicity. Treatment with RGO suppressed the growth of the microalgae. The 72-h IC50 values of RGO evaluated using the logistic and Gompertz models were 148 and 151 mg L(-1), respectively. RGO significantly inhibited Chl a and Chl a/b levels in the algal cells. Chlorophyll a fluorescence analysis showed that RGO significantly down-regulated photosystem II activity. The mechanism of how RGO inhibited algal growth and photosynthetic performance was determined by analyzing the alterations in ultrastructural morphology. RGO adhered to the algal cell surface as a semitranslucent coating. Cell wall damage and membrane integrity loss occurred in the treated cells. ...
Quadratic programming problems arise frequently in various industrial application settings, inclu... more Quadratic programming problems arise frequently in various industrial application settings, including facility location, multiperiod refinery, and circle packing problems. The general nonconvex quadratically constrained quadratic programming (QCQP) problem is NP hard and presents a significant challenge. We present a tight linear relaxation scheme for QCQP that can be used in the context of a branch-and-bound global optimization algorithm for this problem. Most of the current approaches rely on convex relaxations of the QCQP problem which are either nonlinear [7, 2] or relaxing each nonconvex term separately [1, 5]. The main idea of our approach is to relax the entire quadratic constraints by linear underestimations. We propose a relaxation that is tighter than standard linear relaxation based on the convex envelope of multilinear functions [4, 3, 6] and develop an efficient mechanism to tighten the standard relaxation via a class of multilinear cutting planes. Computational experie...
Sepsis, a clinical syndrome occurring in patients following infection or injury, is a leading cau... more Sepsis, a clinical syndrome occurring in patients following infection or injury, is a leading cause of morbidity and mortality worldwide. CD86 (B7-2) is a costimulatory molecule on antigen-presenting cells and plays critical roles in immune responses. In the current study, we investigated the association of two CD86 polymorphisms, rs1129055G/A and rs17281995G/C, with susceptibility to pneumonia-induced sepsis and examined the effects of these two polymorphisms on gene expression in monocytes. CD86 rs1129055G/A and rs17281995G/C were identified in 192 pneumonia-induced septic patients and 201 healthy controls. Data showed that frequencies of the rs1129055GA and AA genotypes were significantly lower in patients than in controls (odds ratio [OR] = 0.57, 95 % confidence interval [CI], 0.35-0.93, p = 0.023, and OR = 0.40, 95 % CI, 0.23-0.71, p = 0.002). Interestingly, the other polymorphism, rs17281995G/C, revealed significantly increased numbers in pneumonia-induced sepsis compared to c...
... Dual bounds and optimality cuts for all-quadratic programs with convex constraints. ... Exact... more ... Dual bounds and optimality cuts for all-quadratic programs with convex constraints. ... Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations. ... linearly on the vector x i provided that all the remaining n−1 vector arguments are fixed. ...
At the intersection of nonlinear and combinatorial optimization, quadratic programming has attrac... more At the intersection of nonlinear and combinatorial optimization, quadratic programming has attracted significant interest over the past several decades. A variety of relaxations for quadratically constrained quadratic programming (QCQP) can be formulated as semidefinite programs (SDPs). The primary purpose of this paper is to present a systematic comparison of SDP relaxations for QCQP. Using theoretical analysis, it is shown that the recently developed doubly nonnegative relaxation is equivalent to the Shor relaxation, when the latter is enhanced with a partial first-order relaxation-linearization technique. These two relaxations are shown to theoretically dominate six other SDP relaxations. A computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations, while providing relaxation gaps averaging 3% as opposed to gaps of up to 19% for weaker relaxations, on 700 randomly generated problems with up to 60 variables. An SDP relaxation derived from Lagrangian relaxation, after the addition of redundant nonlinear constraints to the primal, achieves gaps averaging 13% in a few CPU seconds.
Quadratic programming problems arise frequently in various industrial application settings, inclu... more Quadratic programming problems arise frequently in various industrial application settings, including facility location, multiperiod refinery, and circle packing problems. The general nonconvex quadratically constrained quadratic programming (QCQP) problem is NP hard and presents a significant challenge. We present a tight linear relaxation scheme for QCQP that can be used in the context of a branch-and-bound global optimization algorithm for this problem. Most of the current approaches rely on convex relaxations of the QCQP problem which are either nonlinear [7, 2] or relaxing each nonconvex term separately [1, 5]. The main idea of our approach is to relax the entire quadratic constraints by linear underestimations. We propose a relaxation that is tighter than standard linear relaxation based on the convex envelope of multilinear functions [4, 3, 6] and develop an efficient mechanism to tighten the standard relaxation via a class of multilinear cutting planes. Computational experie...
... Dual bounds and optimality cuts for all-quadratic programs with convex constraints. ... Exact... more ... Dual bounds and optimality cuts for all-quadratic programs with convex constraints. ... Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations. ... linearly on the vector x i provided that all the remaining n−1 vector arguments are fixed. ...
At the intersection of nonlinear and combinatorial optimization, quadratic programming has attrac... more At the intersection of nonlinear and combinatorial optimization, quadratic programming has attracted significant interest over the past several decades. A variety of relaxations for quadratically constrained quadratic programming (QCQP) can be formulated as semidefinite programs (SDPs). The primary purpose of this paper is to present a systematic comparison of SDP relaxations for QCQP. Using theoretical analysis, it is shown that the recently developed doubly nonnegative relaxation is equivalent to the Shor relaxation, when the latter is enhanced with a partial first-order relaxation-linearization technique. These two relaxations are shown to theoretically dominate six other SDP relaxations. A computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations, while providing relaxation gaps averaging 3% as opposed to gaps of up to 19% for weaker relaxations, on 700 randomly generated problems with up to 60 variables. An SDP relaxation derived from Lagrangian relaxation, after the addition of redundant nonlinear constraints to the primal, achieves gaps averaging 13% in a few CPU seconds.
We consider global optimization of nonconvex problems containing multilinear functions. It is wel... more We consider global optimization of nonconvex problems containing multilinear functions. It is well known that the convex hull of a multilinear function over a box is polyhedral, and the facets of this polyhedron can be obtained by solving a linear optimization problem (LP). When used as cutting planes, these facets can significantly enhance the quality of conventional relaxations in general-purpose global solvers. However, in general, the size of this LP grows exponentially with the number of variables in the multilinear function. To cope with this growth, we propose a graph decomposition scheme that exploits the structure of a multilinear function to decompose it to lower-dimensional components, for which the aforementioned LP can be solved very efficiently by employing a customized simplex algorithm. We embed this cutting plane generation strategy at every node of the branch-and-reduce global solver BARON, and carry out an extensive computational study on quadratically constrained quadratic problems, multilinear problems, and polynomial optimization problems.
Increased use of graphene materials might ultimately lead to their release into the environment. ... more Increased use of graphene materials might ultimately lead to their release into the environment. However, only a few studies have investigated the impact of graphene-based materials on green plants. In this study, the impact of reduced graphene oxide (RGO) on the microalgae Scenedesmus obliquus was evaluated to determine its phytotoxicity. Treatment with RGO suppressed the growth of the microalgae. The 72-h IC50 values of RGO evaluated using the logistic and Gompertz models were 148 and 151 mg L(-1), respectively. RGO significantly inhibited Chl a and Chl a/b levels in the algal cells. Chlorophyll a fluorescence analysis showed that RGO significantly down-regulated photosystem II activity. The mechanism of how RGO inhibited algal growth and photosynthetic performance was determined by analyzing the alterations in ultrastructural morphology. RGO adhered to the algal cell surface as a semitranslucent coating. Cell wall damage and membrane integrity loss occurred in the treated cells. ...
Quadratic programming problems arise frequently in various industrial application settings, inclu... more Quadratic programming problems arise frequently in various industrial application settings, including facility location, multiperiod refinery, and circle packing problems. The general nonconvex quadratically constrained quadratic programming (QCQP) problem is NP hard and presents a significant challenge. We present a tight linear relaxation scheme for QCQP that can be used in the context of a branch-and-bound global optimization algorithm for this problem. Most of the current approaches rely on convex relaxations of the QCQP problem which are either nonlinear [7, 2] or relaxing each nonconvex term separately [1, 5]. The main idea of our approach is to relax the entire quadratic constraints by linear underestimations. We propose a relaxation that is tighter than standard linear relaxation based on the convex envelope of multilinear functions [4, 3, 6] and develop an efficient mechanism to tighten the standard relaxation via a class of multilinear cutting planes. Computational experie...
Sepsis, a clinical syndrome occurring in patients following infection or injury, is a leading cau... more Sepsis, a clinical syndrome occurring in patients following infection or injury, is a leading cause of morbidity and mortality worldwide. CD86 (B7-2) is a costimulatory molecule on antigen-presenting cells and plays critical roles in immune responses. In the current study, we investigated the association of two CD86 polymorphisms, rs1129055G/A and rs17281995G/C, with susceptibility to pneumonia-induced sepsis and examined the effects of these two polymorphisms on gene expression in monocytes. CD86 rs1129055G/A and rs17281995G/C were identified in 192 pneumonia-induced septic patients and 201 healthy controls. Data showed that frequencies of the rs1129055GA and AA genotypes were significantly lower in patients than in controls (odds ratio [OR] = 0.57, 95 % confidence interval [CI], 0.35-0.93, p = 0.023, and OR = 0.40, 95 % CI, 0.23-0.71, p = 0.002). Interestingly, the other polymorphism, rs17281995G/C, revealed significantly increased numbers in pneumonia-induced sepsis compared to c...
... Dual bounds and optimality cuts for all-quadratic programs with convex constraints. ... Exact... more ... Dual bounds and optimality cuts for all-quadratic programs with convex constraints. ... Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations. ... linearly on the vector x i provided that all the remaining n−1 vector arguments are fixed. ...
At the intersection of nonlinear and combinatorial optimization, quadratic programming has attrac... more At the intersection of nonlinear and combinatorial optimization, quadratic programming has attracted significant interest over the past several decades. A variety of relaxations for quadratically constrained quadratic programming (QCQP) can be formulated as semidefinite programs (SDPs). The primary purpose of this paper is to present a systematic comparison of SDP relaxations for QCQP. Using theoretical analysis, it is shown that the recently developed doubly nonnegative relaxation is equivalent to the Shor relaxation, when the latter is enhanced with a partial first-order relaxation-linearization technique. These two relaxations are shown to theoretically dominate six other SDP relaxations. A computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations, while providing relaxation gaps averaging 3% as opposed to gaps of up to 19% for weaker relaxations, on 700 randomly generated problems with up to 60 variables. An SDP relaxation derived from Lagrangian relaxation, after the addition of redundant nonlinear constraints to the primal, achieves gaps averaging 13% in a few CPU seconds.
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