An extended supporting hyperplane algorithm for convex MINLP problems

Transcription

1 An extended supporting hyperplane algorithm for convex MINLP problems Jan Kronqvist, Andreas Lundell and Tapio Westerlund Center of Excellence in Optimization and Systems Engineering Åbo Akademi University, Finland OSE annual seminar Turku November 14, 2014

2 2 37 Contents of the talk The extended cutting plane (ECP) algorithm is briefly introduced.

3 2 37 Contents of the talk The extended cutting plane (ECP) algorithm is briefly introduced. The extended supporting hyperplane (ESH) algorithm, a new algorithm for solving convex MINLP problems to global optimality, is introduced Cutting planes are replaced with supporting hyperplanes using a line search procedure. Two LP preprocessing steps are utilized to quickly get a tight linear relaxation of the part of the feasible region defined by the convex/quasiconvex constraints.

4 The ECP algorithm

5 The ECP algorithm 4 37 The extended cutting plane algorithm The ECP algorithm is a solver for generally convex mixed-integer nonlinear programming (MINLP) problems.

6 The ECP algorithm 4 37 The extended cutting plane algorithm The ECP algorithm is a solver for generally convex mixed-integer nonlinear programming (MINLP) problems. Solves MILP relaxations of the MINLP problem where the nonlinear constraints are approximated using cutting planes.

7 The ECP algorithm 4 37 The extended cutting plane algorithm The ECP algorithm is a solver for generally convex mixed-integer nonlinear programming (MINLP) problems. Solves MILP relaxations of the MINLP problem where the nonlinear constraints are approximated using cutting planes. Implemented, e.g., in the AlphaECP solver in GAMS and available on the NEOS server.

8 The ECP algorithm 5 37 The extended cutting plane algorithm First presented in internal report in 1992 Inspired by Kelley s cutting plane method Convex NLP problems Kelley Jr. J., The cutting-plane method for solving convex programs, Journal of the SIAM, vol. 8(4), pp , Published in Computers & Chemical Engineering in 1995 Westerlund T. and Pettersson F., An extended cutting plane method for solving convex MINLP problems, Computers & Chemical Engineering 19, pp , 1995.

9 The ECP algorithm 6 37 Extensions Pseudoconvex constraints Westerlund T., Skrifvars H., Harjunkoski I. and Pörn R. An extended cutting plane method for solving a class of non-convex MINLP problems. Computers and Chemical Engineering, 22, , 1998.

10 The ECP algorithm 6 37 Extensions Pseudoconvex constraints Westerlund T., Skrifvars H., Harjunkoski I. and Pörn R. An extended cutting plane method for solving a class of non-convex MINLP problems. Computers and Chemical Engineering, 22, , Pseudoconvex objective function and constraints Westerlund T. and Pörn R. Solving Pseudo-Convex Mixed Integer Optimization Problems by Cutting Plane techniques. Optimization and Engineering, 3, , 2002.

11 The ECP algorithm 6 37 Extensions Pseudoconvex constraints Westerlund T., Skrifvars H., Harjunkoski I. and Pörn R. An extended cutting plane method for solving a class of non-convex MINLP problems. Computers and Chemical Engineering, 22, , Pseudoconvex objective function and constraints Westerlund T. and Pörn R. Solving Pseudo-Convex Mixed Integer Optimization Problems by Cutting Plane techniques. Optimization and Engineering, 3, , Nonsmooth constraints Eronen V.-P., Mäkelä M. M. and Westerlund T. On the generalization of ECP and OA methods to nonsmooth convex MINLP problems, Taylor and Francis, 2014.

12 The ECP algorithm 7 37 An example minimize c T x = x 1 x 2 subject to g 1 (x 1,x 2 ) = 0.15(x 1 8) (x 2 6) e x 1x g 2 (x 1,x 2 ) = 1/x 1 + 1/x2 x x x 1 3x x 1 20, 1 x 2 20, x 1, x x 2 10 x x x 1

13 The ECP algorithm 7 37 An example minimize c T x = x 1 x 2 subject to g 1 (x 1,x 2 ) = 0.15(x 1 8) (x 2 6) e x 1x g 2 (x 1,x 2 ) = 1/x 1 + 1/x2 x x x 1 3x x 1 20, 1 x 2 20, x 1, x x 2 10 x x x 1

14 The ECP algorithm 7 37 An example minimize c T x = x 1 x 2 subject to g 1 (x 1,x 2 ) = 0.15(x 1 8) (x 2 6) e x 1x g 2 (x 1,x 2 ) = 1/x 1 + 1/x2 x x x 1 3x x 1 20, 1 x 2 20, x 1, x x 2 10 x x x 1

15 The ECP algorithm 7 37 An example minimize c T x = x 1 x 2 subject to g 1 (x 1,x 2 ) = 0.15(x 1 8) (x 2 6) e x 1x g 2 (x 1,x 2 ) = 1/x 1 + 1/x2 x x x 1 3x x 1 20, 1 x 2 20, x 1, x x 2 10 x x x 1

16 The ECP algorithm 8 37 In each iteration k of the algorithm a MILP problem is solved to obtain the solution point (x k 1,xk 2 ). First iteration gives x 1 1 = 20,x1 2 = 20.

17 The ECP algorithm 8 37 In each iteration k of the algorithm a MILP problem is solved to obtain the solution point (x k 1,xk 2 ). First iteration gives x1 1 = 20,x1 2 = 20. g 1 (x1 1,x1 2 ) = , g 2(x1 1,x1 2 ) = 15.9.

18 The ECP algorithm 8 37 In each iteration k of the algorithm a MILP problem is solved to obtain the solution point (x k 1,xk 2 ). First iteration gives x1 1 = 20,x1 2 = 20. g 1 (x1 1,x1 2 ) = , g 2(x1 1,x1 2 ) = A new cutting plane is generated for the violated nonlinear constraint g 1 : g 1 (x 1 1,x1 2 ) + g 1(x 1 1,x1 2 )T (x x 1 1,x x1 2 ) 0

19 The ECP algorithm 9 37 The nonlinear function g 1 (x 1,x 2 )

20 The ECP algorithm The nonlinear function g 1 (x 1,x 2 ) and the cutting plane generated at x 1 = 20,x 2 = 20

21 The ECP algorithm To solve the problem with the ECP algorithm ( = 0.001) it takes 17 iterations (17 MILP problems solved to optimality).

22 The ECP algorithm To solve the problem with the ECP algorithm ( = 0.001) it takes 17 iterations (17 MILP problems solved to optimality). How can we improve performance?

23 The ECP algorithm To solve the problem with the ECP algorithm ( = 0.001) it takes 17 iterations (17 MILP problems solved to optimality). How can we improve performance? Generate cutting planes on the boundary of the feasible set!

24 The ESH algorithm

25 The ESH algorithm A new interior point based algorithm for solving convex MINLP problems to global optimality.

26 The ESH algorithm A new interior point based algorithm for solving convex MINLP problems to global optimality. Roots: The extended cutting plane algorithm The supporting hyperplane method for unimodal programming, Veinott Jr. A. F., Operations Research, Vol. 15(1), pp , 1967.

27 The ESH algorithm A new interior point based algorithm for solving convex MINLP problems to global optimality. Roots: The extended cutting plane algorithm 1995 Kelley s cutting plane algorithm The supporting hyperplane method for unimodal programming, Veinott Jr. A. F., Operations Research, Vol. 15(1), pp , 1967.

28 The ESH algorithm A new interior point based algorithm for solving convex MINLP problems to global optimality. Roots: The extended cutting plane algorithm 1995 Kelley s cutting plane algorithm 1960 The supporting hyperplane method The supporting hyperplane method for unimodal programming, Veinott Jr. A. F., Operations Research, Vol. 15(1), pp , 1967.

29 The ESH algorithm A new interior point based algorithm for solving convex MINLP problems to global optimality. Roots: The extended cutting plane algorithm 1995 Kelley s cutting plane algorithm 1960 The supporting hyperplane method Cutting planes are replaced with supporting hyperplanes using a line search procedure to find the generation point. An interior point is required for the line search. 1 The supporting hyperplane method for unimodal programming, Veinott Jr. A. F., Operations Research, Vol. 15(1), pp , 1967.

30 The ESH algorithm A new interior point based algorithm for solving convex MINLP problems to global optimality. Roots: The extended cutting plane algorithm 1995 Kelley s cutting plane algorithm 1960 The supporting hyperplane method Cutting planes are replaced with supporting hyperplanes using a line search procedure to find the generation point. An interior point is required for the line search. Two LP preprocessing steps are utilized to quickly get a tight linear relaxation of the part of the feasible region defined by the convex/quasiconvex constraints. 1 The supporting hyperplane method for unimodal programming, Veinott Jr. A. F., Operations Research, Vol. 15(1), pp , 1967.

31 The ESH algorithm The MINLP problem The algorithm finds the optimal solution x to the following convex MINLP problem: x = argmin c T x x C L Y (P) where x = [x 1,x 2,...,x N ] T belongs to the compact set X = { x xi x i x i, i = 1,...,N } n, the feasible region is defined by C L Y, C = {x g m (x) 0, m = 1,...,M, x X }, L = {x Ax a, Bx = b, x X }, Y = {x x i, i I, x X }, and C is a convex set.

32 The ESH algorithm Steps in the ESH algorithm NLP: Obtain a feasible, relaxed interior point (a point in the set C) by solving a NLP problem.

33 The ESH algorithm Steps in the ESH algorithm NLP: Obtain a feasible, relaxed interior point (a point in the set C) by solving a NLP problem. LP1: Solve simple LP problems (initially in X) to add additional supporting hyperplanes.

34 The ESH algorithm Steps in the ESH algorithm NLP: Obtain a feasible, relaxed interior point (a point in the set C) by solving a NLP problem. LP1: Solve simple LP problems (initially in X) to add additional supporting hyperplanes. LP2: Solve simple LP problems (including the constraints in L) to add additional supporting hyperplanes.

35 The ESH algorithm Steps in the ESH algorithm NLP: Obtain a feasible, relaxed interior point (a point in the set C) by solving a NLP problem. LP1: Solve simple LP problems (initially in X) to add additional supporting hyperplanes. LP2: Solve simple LP problems (including the constraints in L) to add additional supporting hyperplanes. MILP: Solve MILP problems to find the optimal solution to (P).

36 The ESH algorithm NLP step If an interior point is not given, obtain a feasible, relaxed interior point (satisfying all the nonlinear constraints in C) by solving a NLP problem

37 The ESH algorithm LP1 step (optional) Solve simple LP problems (initially in X) and conduct a line search procedure to obtain supporting hyperplanes giving a first linear relaxation of the convex set C

38 The ESH algorithm LP1 step (optional) Solve simple LP problems (initially in X) and conduct a line search procedure to obtain supporting hyperplanes giving a first linear relaxation of the convex set C

39 The ESH algorithm LP1 step (optional) Solve simple LP problems (initially in X) and conduct a line search procedure to obtain supporting hyperplanes giving a first linear relaxation of the convex set C

40 The ESH algorithm LP1 step (optional) Solve simple LP problems (initially in X) and conduct a line search procedure to obtain supporting hyperplanes giving a first linear relaxation of the convex set C

41 The ESH algorithm LP1 step (optional) Solve simple LP problems (initially in X) and conduct a line search procedure to obtain supporting hyperplanes giving a first linear relaxation of the convex set C

42 The ESH algorithm LP1 step (optional) Solve simple LP problems (initially in X) and conduct a line search procedure to obtain supporting hyperplanes giving a first linear relaxation of the convex set C

43 The ESH algorithm LP1 step (optional) Solve simple LP problems (initially in X) and conduct a line search procedure to obtain supporting hyperplanes giving a first linear relaxation of the convex set C

44 The ESH algorithm LP1 step (optional) Solve simple LP problems (initially in X) and conduct a line search procedure to obtain supporting hyperplanes giving a first linear relaxation of the convex set C

45 The ESH algorithm LP1 step (optional) Solve simple LP problems (initially in X) and conduct a line search procedure to obtain supporting hyperplanes giving a first linear relaxation of the convex set C

46 The ESH algorithm LP2 step (optional) Continue with a corresponding procedure as in LP1 but now also including the linear constraints in L in the original problem

47 The ESH algorithm LP2 step (optional) Continue with a corresponding procedure as in LP1 but now also including the linear constraints in L in the original problem

48 The ESH algorithm LP2 step (optional) Continue with a corresponding procedure as in LP1 but now also including the linear constraints in L in the original problem

49 The ESH algorithm MILP step Finally include the integer requirements and solve MILP problems using a corresponding procedure to find the optimal solution to (P)

50 The ESH algorithm MILP step Finally include the integer requirements and solve MILP problems using a corresponding procedure to find the optimal solution to (P)

51 The ESH algorithm MILP step Finally include the integer requirements and solve MILP problems using a corresponding procedure to find the optimal solution to (P)

52 The ESH algorithm MILP step Finally include the integer requirements and solve MILP problems using a corresponding procedure to find the optimal solution to (P)

53 The ESH algorithm NLP step A point in C is required as an endpoint for the line searches to be conducted in the LP1-, LP2- and MILP-steps.

54 The ESH algorithm NLP step A point in C is required as an endpoint for the line searches to be conducted in the LP1-, LP2- and MILP-steps. Assuming that (P) has a solution, the internal point can be obtained from the following NLP problem: x NLP = argminf(x), x X where F(x) := max {g m(x)}. m=1,...,m (P-NLP)

55 The ESH algorithm NLP step A point in C is required as an endpoint for the line searches to be conducted in the LP1-, LP2- and MILP-steps. Assuming that (P) has a solution, the internal point can be obtained from the following NLP problem: x NLP = argminf(x), x X where F(x) := max {g m(x)}. m=1,...,m (P-NLP) F is convex/quasiconvex since it is the maximum of convex/quasiconvex functions.

56 The ESH algorithm NLP step A point in C is required as an endpoint for the line searches to be conducted in the LP1-, LP2- and MILP-steps. Assuming that (P) has a solution, the internal point can be obtained from the following NLP problem: x NLP = argminf(x), x X where F(x) := max {g m(x)}. m=1,...,m (P-NLP) F is convex/quasiconvex since it is the maximum of convex/quasiconvex functions. (P-NLP) may be nonsmooth (if M > 1) even if g m is smooth.

57 The ESH algorithm NLP step A point in C is required as an endpoint for the line searches to be conducted in the LP1-, LP2- and MILP-steps. Assuming that (P) has a solution, the internal point can be obtained from the following NLP problem: x NLP = argminf(x), x X where F(x) := max {g m(x)}. m=1,...,m (P-NLP) F is convex/quasiconvex since it is the maximum of convex/quasiconvex functions. (P-NLP) may be nonsmooth (if M > 1) even if g m is smooth. The point x NLP need not be optimal but then fulfill F( x NLP ) < 0.

58 The ESH algorithm NLP step A point in C is required as an endpoint for the line searches to be conducted in the LP1-, LP2- and MILP-steps. Assuming that (P) has a solution, the internal point can be obtained from the following NLP problem: x NLP = argminf(x), x X where F(x) := max {g m(x)}. m=1,...,m (P-NLP) F is convex/quasiconvex since it is the maximum of convex/quasiconvex functions. (P-NLP) may be nonsmooth (if M > 1) even if g m is smooth. The point x NLP need not be optimal but then fulfill F( x NLP ) < 0. The NLP step can also be formulated as a smooth NLP problem if all functions g m are smooth and convex.

59 The ESH algorithm LP1 step Starting from k = 1, 0 = X, the problem x k LP = argmin k 1 c T x (P-LP1) is solved, and the point x k is obtained by a line search for F(x k ) = 0 between the internal point x NLP and the solution point to (P-LP1) x k LP : x k = x NLP + (1 ) x LP k, [0,1].

60 The ESH algorithm LP1 step Starting from k = 1, 0 = X, the problem x k LP = argmin k 1 c T x (P-LP1) is solved, and the point x k is obtained by a line search for F(x k ) = 0 between the internal point x NLP and the solution point to (P-LP1) x k LP : x k = x NLP + (1 ) x LP k, [0,1]. Supporting hyperplanes (SHs) l k := F(x k ) + F (x k ) T (x x k ) 0 are generated and added to k. F (x k ) T is a gradient or subgradient of F at x k.

61 The ESH algorithm LP1 step Starting from k = 1, 0 = X, the problem x k LP = argmin k 1 c T x (P-LP1) is solved, and the point x k is obtained by a line search for F(x k ) = 0 between the internal point x NLP and the solution point to (P-LP1) x k LP : x k = x NLP + (1 ) x LP k, [0,1]. Supporting hyperplanes (SHs) l k := F(x k ) + F (x k ) T (x x k ) 0 are generated and added to k. F (x k ) T is a gradient or subgradient of F at x k. Repeted untill F( x k LP ) < LP1 or a maximum number of SHs

62 The ESH algorithm LP2 step This step is otherwise identical to LP1, with the exception that the linear constraints in L are now also included, i.e., x k LP = argmin k 1 L c T x (P-LP2)

63 The ESH algorithm LP2 step This step is otherwise identical to LP1, with the exception that the linear constraints in L are now also included, i.e., x k LP = argmin k 1 L c T x (P-LP2) (P-LP2) is repeatedly solved until F( x k LP ) < LP2 or a maximum number of SHs have additionally been generated.

64 The ESH algorithm MILP step Finally, in order to also fulfill the integer requirements of problem (P), a MILP step is performed.

65 The ESH algorithm MILP step Finally, in order to also fulfill the integer requirements of problem (P), a MILP step is performed. This step is otherwise identical to LP2, with the exception that the integer requirements in Y are now additionally considered, i.e., x k MILP = argmin k 1 L Y c T x. (P-MILP)

66 The ESH algorithm MILP step Finally, in order to also fulfill the integer requirements of problem (P), a MILP step is performed. This step is otherwise identical to LP2, with the exception that the integer requirements in Y are now additionally considered, i.e., x k MILP = argmin k 1 L Y c T x. (P-MILP) (P-MILP) is repeatedly solved until F( x k MILP ) < MILP.

67 The ESH algorithm MILP step Finally, in order to also fulfill the integer requirements of problem (P), a MILP step is performed. This step is otherwise identical to LP2, with the exception that the integer requirements in Y are now additionally considered, i.e., x k MILP = argmin k 1 L Y c T x. (P-MILP) (P-MILP) is repeatedly solved until F( x k MILP ) < MILP. Intermediate (P-MILP) problems do not need to be solved to optimality, but in order to guarantee an optimal solution of (P), the final MILP solution must be optimal.

68 The ESH algorithm Now, consider the same example as earlier minimize c T x = x 1 x 2 subject to 0.15(x 1 8) (x 2 6) e x 1x /x 1 + 1/x2 x x x 1 3x x 1 20, 1 x 2 20, x 1, x x 2 10 x x x 1

69 The ESH algorithm NLP step find an interior point x NLP = argmin F(x 1,x 2 ), (x 1,x 2 ) X where F(x 1,x 2 ) := max{g 1 (x 1,x 2 ), g 2 (x 1,x 2 )}. The problem can be found using a suitable NLP solver. Not required to be the optimal point The optimal point here is (7.45, 8.54)

70 The ESH algorithm LP1-step Iteration 1 Assume initially that 0 = X

71 The ESH algorithm LP1-step Iteration 1 Assume initially that 0 = X. k = 1, solve LP in, x k LP = argmin k 1 c T x

72 The ESH algorithm LP1-step Iteration 1 Assume initially that 0 = X. k = 1, solve LP in, x k LP = argmin k 1 c T x Do line search x k = x NLP + (1 ) x k LP.

73 The ESH algorithm LP1-step Iteration 1 Assume initially that 0 = X. k = 1, solve LP in, x k LP = argmin k 1 c T x Do line search x k = x NLP + (1 ) x k LP. Generate supporting hyperplane in x k and add to.

74 The ESH algorithm LP1-step Iteration 2 1 = {x l 1 (x) 0, x X}. l 1 (x) = 3.26x x

75 The ESH algorithm LP1-step Iteration 2 1 = {x l 1 (x) 0, x X}. l 1 (x) = 3.26x x k = 2, solve LP in, x k LP = argmin k 1 c T x

76 The ESH algorithm LP1-step Iteration 2 1 = {x l 1 (x) 0, x X}. l 1 (x) = 3.26x x k = 2, solve LP in, x k LP = argmin k 1 c T x Do line search x k = x NLP + (1 ) x k LP.

77 The ESH algorithm LP1-step Iteration 2 1 = {x l 1 (x) 0, x X}. l 1 (x) = 3.26x x k = 2, solve LP in, x k LP = argmin k 1 c T x Do line search x k = x NLP + (1 ) x k LP. Generate supporting hyperplane in x k and add to.

78 The ESH algorithm LP1-step Iteration 3 2 = {x l j (x) 0, j {1,2}, x X} l 1 (x) = 3.26x x l 2 (x) = 0.332x x

79 The ESH algorithm LP1-step Iteration 3 2 = {x l j (x) 0, j {1,2}, x X} l 1 (x) = 3.26x x l 2 (x) = 0.332x x k = 3, solve LP in, x k LP = argmin k 1 c T x.

80 The ESH algorithm LP1-step Iteration 3 2 = {x l j (x) 0, j {1,2}, x X} l 1 (x) = 3.26x x l 2 (x) = 0.332x x k = 3, solve LP in, x k LP = argmin k 1 c T x. Do line search, generate supporting hyperplane and add to.

81 The ESH algorithm LP1-step Iteration 3 2 = {x l j (x) 0, j {1,2}, x X} l 1 (x) = 3.26x x l 2 (x) = 0.332x x k = 3, solve LP in, x k LP = argmin k 1 c T x. Do line search, generate supporting hyperplane and add to. Terminate LP1-step since F( x k LP ) < LP1.

82 The ESH algorithm LP2-step Iteration 4 3 = {x l j (x) 0, j {1,2,3}, x X} l 1 (x) = 3.26x x l 2 (x) = 0.332x x l 3 (x) = 1.66x x

83 The ESH algorithm LP2-step Iteration 4 3 = {x l j (x) 0, j {1,2,3}, x X} l 1 (x) = 3.26x x l 2 (x) = 0.332x x l 3 (x) = 1.66x x k = 4, solve LP now in L, x k LP = argmin k 1 L c T x.

84 The ESH algorithm LP2-step Iteration 4 3 = {x l j (x) 0, j {1,2,3}, x X} l 1 (x) = 3.26x x l 2 (x) = 0.332x x l 3 (x) = 1.66x x k = 4, solve LP now in L, x k LP = argmin k 1 L c T x. Do line search, generate supporting hyperplane and add to.

85 The ESH algorithm LP2-step Iteration 4 3 = {x l j (x) 0, j {1,2,3}, x X} l 1 (x) = 3.26x x l 2 (x) = 0.332x x l 3 (x) = 1.66x x k = 4, solve LP now in L, x LP k = argmin k 1 L c T x. Do line search, generate supporting hyperplane and add to. Terminate LP2-step since F( x LP k ) < LP2.

86 The ESH algorithm MILP step Iterations 5 and 6 20 MILP k = 5 20 MILP k = In this step the integer requirements in Y are also considered, i.e., initially k = 5, = k 1 L Y. The MILP steps are required to guarantee an integer-feasible solution.

87 The ESH algorithm Solution and comparisons to other solvers Solving the MINLP problem with the supporting hyperplane algorithm gives the following solution Type Iteration Obj. funct. x 1 x 2 F(x 1,x 2 ) LP LP LP LP MILP MILP

88 The ESH algorithm Solution and comparisons to other solvers Solving the MINLP problem with the supporting hyperplane algorithm gives the following solution Type Iteration Obj. funct. x 1 x 2 F(x 1,x 2 ) LP LP LP LP MILP MILP Solution times compared to some other MINLP solvers: Solver Subproblems solved Time (s) Implementation ESH 1 NLP + 4 LP + 2 MILP 0.04 Early prototype ECP 21 MILP (10 OPT) + 1 NLP 1.4 GAMS CPLEX DICOPT 10 NLP + 10 MILP 1.5 GAMS CONOPT + CPLEX

89 The ESH algorithm Solver comparison The test problems are taken from MINLPlib2, which is a collection Mixed Integer Nonlinear Programming models. Solver fo7-ar4-1 fo9-ar3-1 jit1 m7-ar5-1 ESH ECP ANTIGONE BARON DICOPT # # # # SBB # # 0.03 # SCIP Variables Binaries Integers Type MINLP MINLP MINLP MINLP

90 The ESH algorithm Solver comparison Solver batchs101006m enpro56pb o7 rsyn0805m04h rsyn0830m04h sssd25-08 ESH ECP ANTIGONE BARON DICOPT # SBB # # # # SCIP Variables ,400 1, Binaries Integers Type MBNLP MBNLP MBNLP MBNLP MBNLP MBNLP

91 The ESH algorithm Solver comparison Solver alan fac3 netmod-dol2 netmod-kar1 slay05h ESH ECP ANTIGONE BARON # 1, DICOPT # # 0.19 SBB # SCIP Variables , Binaries Integers Type MBQP MBQP MBQP MBQP MBQP

92 The ESH algorithm Solver du-opt ex4 ESH ECP ANTIGONE 0.22 BARON DICOPT # 0.44 SBB SCIP Variables Binaries 0 25 Integers 13 0 Type MIQP MBQCQP Solver comparison

93 The ESH algorithm Future work Implementations of the algorithm Mathematica / Wolfram Language. Early prototype available. COIN-OR: Utilize the Optimization Services and Open Solver Interface APIs. GAMS

94 The ESH algorithm Future work Implementations of the algorithm Mathematica / Wolfram Language. Early prototype available. COIN-OR: Utilize the Optimization Services and Open Solver Interface APIs. GAMS Development of the algorithm Pseudoconvex constraints and objective functions. Selection (update) strategies of the interior point. Strategies for the LP1/LP2 steps.

95 The ESH algorithm Thank you!