ANTIGONE: Algorithms for coNTinuous / Integer Global Optimization of ...

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ANTIGONE: Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations Ruth Misener · Christodoulos A. Floudas

Received: date / Accepted: date

Abstract This manuscript introduces ANTIGONE, Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations, a general mixed-integer nonlinear global optimization framework. ANTIGONE is the evolution of the Global Mixed-Integer Quadratic Optimizer, GloMIQO, to general nonconvex terms. The purpose of this paper is to show how the extensible structure of ANTIGONE realizes our previously-proposed mixed-integer quadratically-constrained quadratic program (MIQCQP) and mixed-integer signomial optimization (MISO) computational frameworks. To demonstrate the capacity of ANTIGONE, this paper presents computational results on a test suite of 2571 problems from standard libraries and the open literature; we compare ANTIGONE to other state-of-the-art global optimization solvers. Keywords MINLP; deterministic global optimization; optimization software; branch-andcut C.A.F. is thankful for support from the National Science Foundation (CBET – 0827907). This material is based upon work supported by both the National Science Foundation Graduate Research Fellowship to R.M. under Grant No. DGE-0646086 and by the Royal Academy of Engineering Research Fellowship to R.M. R. Misener · C. A. Floudas Department of Chemical and Biological Engineering Princeton University Princeton, NJ 08544-5263; USA Tel.: +1-609-258-4595 Fax: +1-609-258-0211 E-mail: [email protected] Present address: of R. Misener Department of Chemical Engineering Imperial College London South Kensington SW7 2AZ; UK

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Ruth Misener, Christodoulos A. Floudas

1 Introduction Deterministic global optimization of mixed-integer nonlinear programs (MINLP) is broadly applicable in diverse domains ranging from molecular biology to refinery operations to computational chemistry to synthesizing sustainable processes [17, 24, 50–52, 54, 63, 64, 101, 103, 136]. Our recent work developed solution strategies for two classes of deterministic global optimization problems: mixed-integer quadratically constrained quadratic programs (MIQCQP) and mixed-integer signomial optimization problems (MISO) [102, 104, 105, 107]. A software implementation illustrated and validated the proposed MIQCQP and MISO computational frameworks; we further collaborated with the GAMS Development Corporation to publicly release the MIQCQP framework as the Global Mixed-Integer Quadratic Optimizer (GloMIQO; first available in GAMS 23.8). GloMIQO has been subsequently used for applications including: optimal crystal orientation [62]; multiperiod blend scheduling [75, 76]; network flow [131]; water network design [35, 73]; cutting ellipses from areaminimizing rectangles [69]. This manuscript presents ANTIGONE (Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Expressions; first available in GAMS 24.1), a computational framework and associated software implementation for globally optimizing nonconvex MINLP. ANTIGONE integrates our previous MIQCQP [102, 104, 107] and MISO [105] work into a cohesive whole; ANTIGONE further generalizes the computational framework to include exponential and logarithmic functions. The purpose of this paper is to show how the extensible structure of ANTIGONE realizes the MIQCQP and MISO computational frameworks [102, 104, 105, 107]. Section 2 defines MINLP and mentions several MINLP global optimization solvers. We develop ANTIGONE in Section 3 as a global optimization framework exploiting an array of special mathematical structure components. Section 3 may be read as a type of critical literature review; the intellectual contribution is the way ANTIGONE ties together its many constitutive algorithms. After defining a test suite of 2571 optimization problems from standard libraries and the open literature in Section 4, we present extensive computational results in Section 5 comparing ANTIGONE to other state-of-the-art global optimization solvers. We conclude in Section 6.

2 Problem Definition MINLP is defined: min

f0 (x, y, z)

UP s.t. bLO m  f m (x, y, z)  bm

8 m 2 {1, . . . , M}

x 2 RC ; y 2 {0, 1}B ; z 2 ZI

(MINLP)

ANTIGONE

3

where C, B, I, and M represent the number of continuous variables, binary variables, integer variables, and constraints, respectively. Parameters vectors bLO and bUP m bound the ⇥ L Um⇤ constraints. We assume that it is possible to infer finite bounds xi , xi on the variables participating in nonlinear terms fm and that the image of fm is finite on x. Typical expressions for f0 (x, y, z) and fm (x, y, z) are:

fm (x, y, z) = cm + aTm (x, y, z) + (x, y, z)T Qm (x, y, z) Sm

C

s=1

c=1

p

Em

Lm

e=1

`=1

+ Â csm · ’ xc sm , c + Â cem · ex + Â c`m · log x where the powers psm , c are constant reals; cm , am , Qm , csm , cem , c`m are constant coefficients; (x, y, z) is a combined vector of variables; Sm , Em , Lm are the number of signomial, exponential, and logarithmic terms, respectively. This manuscript refers exclusively to the functional forms fm presented in Table 1, but note that the ANTIGONE software implementation addresses a wider set of terms through preprocessing transformations. For example, ANTIGONE transforms logarithms of base b into natural logarithms using the transformation logb (x) = log(x)/ log(b) and changes powers with constant base b into exponents (bx = elog(b)·x ). Global optimization of MINLP is a topic of active research. We therefore limit ourselves to mentioning several excellent review articles and texts [17, 24, 40, 49–52, 54, 63, 64, 101, 103, 113, 136, 145] and hereafter discuss the items most relevant to ANTIGONE. The C++ implementation of ANTIGONE belongs to the class of numerical optimization software addressing MINLP; refer to the review of Bussieck and Vigerske [26] for complete treatment. The existing numerical solvers most relevant to ANTIGONE are: aBB [4, 5, 10, 50, 93]; BARON [136–138]; Couenne [16, 88]; GloMIQO [102, 104, 107]; LINDO [58, 85]; SCIP [1, 3, 18, 19, 145]; SGO [89–92]. ANTIGONE performs equivalently to GloMIQO [102, 104, 107] when all nonlinearities in MINLP are quadratic.

3 Global optimization Framework The overarching structure of the ANTIGONE framework is diagrammed in Figure 1; Figure 2 illustrates the inheritance structure of four major ANTIGONE base classes. Sections 3.1 – 3.3 describe the component parts of: reformulating input; finding special mathematical structure; branch-and-cut global optimization.

3.1 Reformulating User Input Like aBB [4, 5, 10, 50, 93] and SGO [89–92], ANTIGONE emphasizes term-based underestimators [105]. ANTIGONE therefore begins by reformulating the user-defined MINLP

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Ruth Misener, Christodoulos A. Floudas

Fig. 1: Given user-defined MINLP, the framework reformulates the model, detects special mathematical structure, solves the optimization problem, and returns the model with respect to the original variables

so that each equation is expressed as a sum of the terms outlined in Table 1. As mentioned in Section 2, ANTIGONE will equivalently rewrite the bases of powers and logarithms to reach the terms in Table 1. For more complex terms, ANTIGONE introduces auxiliary variables [16, 57, 130, 136–138, 145]; this strategy is in contrast to systems exclusively using underestimators that do not introduce auxiliary variables [4, 5, 92, 95, 108]. Figure 2(a) diagrams the inheritance structure of base class term; this implementation is designed in recognition of an accelerating trend towards developing good underestimators for very specific terms (e.g., [32, 61, 71, 72, 83, 86, 89, 91, 93, 94, 98–100, 116, 129, 132–136, 139]). Every term needs to know its: participating variable(s); regions of convexity/concavity and edge-convexity/edge-concavity; partial derivatives; etc. The polymorphic strategy illustrated in Figure 2(a) allows us to easily integrate many disparate term types into a single framework; we also avoid duplicating code for similar types. Observe that specialized terms which are not at this writing integrated into the framework (e.g., [61, 71, 72, 86, 129, 139]) could be easily added to Figure 2(a); we anticipate further advances in the area of term-specific underestimators and have architected ANTIGONE accordingly. The only three term types in Figure 2(a) not analyzed in the open literature are the composite univariate terms (exponential; fractional; logarithmic). These composite terms regularly appear in standard libraries: GLOBALLib (e.g., arki0018, chem, ex6 1 1, ex6 1 2, ex14 1 8, filter); MINLPLib (contvar); AMPL Book Library (nltrans); PrincetonLib (e.g., bigbank, s367, s377). ANTIGONE knows analytic expressions for the second derivatives of these special composite terms and performs interval arithmetic on

ANTIGONE

5

Table 1: Each derived class of base class Term is associated with a specific mathematical expression and several relaxation strategies Term Type

Mathematical Form

Relaxation Strategies

Constant

c

–

Bilinear / Quadratic

x1 · x2

McCormick Hull Outer Approximation

[8, 95] [45, 46, 49, 59]

Multivariate Signomial

x1a1 · x2a2 · · · xnan

Edge-Concave Exponential Transformation Fractional Outer Approximation

[100, 102, 132, 133] [89–91, 94] [116, 135] [59, 93]

Absolute Value

|a · x + b|

Equivalent MILP Representation

[105]

Exponential

ea·x+b

Secant Line Outer Approximation

Linear

a·x+b

–

Logarithmic

log(a · x + b)

Secant Line Outer Approximation

xa

Odd Degree Monomial Secant Line Outer Approximation

Univariate Signomial Composite Exponent Fractional Logarithm

(a1 · x + b1 ) · ea2 ·x+b2 a1 ·x+b1 a2 ·x+b2

[45, 46, 49, 59]

[45, 46, 49, 59] [83] [45, 46, 49, 59]

Secant Line Outer Approximation

[45, 46, 49, 59]

(a1 x + b1 ) log(a2 x + b2 )

d 2 f /dx2 to determine regions of convexity/concavity: 8 > fe (x) = (a1 · x + b1 ) · ea2 ·x+b2 Exponential < + Composite > : d 2 fe = (a1 · a2 · a2 · x + 2 · a1 · a2 + a2 · a2 · b1 ) · ea2 ·x+b2 8 dx2 a1 ·x+b1 > > f f (x) = a2 ·x+b2 Fractional < + Composite > > : d 2 f f = 2 · a · a1 ·b2 a2 ·b1 2 (a ·x+b )3 2 2 8 dx2 > f (x) = (a · x + b1 ) · log(a2 · x + b2 ) > 1 ` < Logarithmic + Composite > > : d 2 f2` = a1 ·a2 ·(a2 ·x+b2 )+a1 ·a22·b2 a2 ·a2 ·b1 dx

(a2 ·x+b2 )

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Ruth Misener, Christodoulos A. Floudas

(a) Term

(b) Expression

(c) Upper Bounding Solver

(d) Variable

Fig. 2: Inheritance Structure of Four ANTIGONE Base Classes: (a) Term; (b) Expression is a list of pointers to Term objects with associated coefficients; (c) Upper Bounding Solver is the NLP solver interface; (d) Variable

After transforming the initial user-defined MINLP to an equivalent MINLP that exclusively uses the terms in Table 1, ANTIGONE performs all of the GloMIQO [102, 104, 107] and the MISO framework [105] reformulations. The only major additions to the reformulation strategies specifically address exponential and logarithmic functions. As in the MISO frame-

ANTIGONE

7

work [105], one goal of reformulation is to flatten the expression tree (i.e., find equivalent functional forms with fewer layers of auxiliary variables). ANTIGONE eliminates a level from a typical factorable programming tree with each of the following two reformulations: ⇢ Flatten Logarithms log(x1a · x2b ) ) a · log x1 + b · log x2 ⇢ (1) Flatten Exponentials ex1 · ex2 ) ex1 +x2

3.2 Elucidating Special Mathematical Structure After reformulating the user-defined MINLP, ANTIGONE detects special mathematical structure that it will exploit in the branch-and-cut phase (Section 3.3). The types of special structure that ANTIGONE considers are: reformulation-linearization technique (RLT) equations; convexity/concavity; edge-convexity/edge-concavity; aBB relaxations; term-specific underestimators. Our previous work has shown how to find and use these components [102, 104, 105, 107]. To summarize: – RLT multiplies every pairwise combination of: variables; nonlinear terms; equations [14, 84, 104, 105, 107, 124–128]. ANTIGONE saves the combinations that do not introduce new terms into the model formulation and updates these equations at every node of the branch-and-cut tree. Special RLT equations are added directly to the model formulation; other RLT equations are used as cutting planes and integrated into the feasibilitybased bounds tightening routines [102, 104, 105, 107]. – Convexity/Concavity permits the easy generation of a cutting plane at a point x: ˆ f (x)

f (x) ˆ + f 0 (x) · (x

f (x)  f (x) ˆ + f 0 (x) · (x

x) ˆ (convex) x) ˆ (concave)

Based on interval arithmetic, terms and multi-term expressions are labeled as always/ sometimes/never convex/concave; this information is used in the branch-and-cut phase [105, 107]. – Edge-Convexity/Edge-Concavity implies a vertex polyhedral envelope; ANTIGONE labels terms and multi-term expressions as always/sometimes/never edge-convex/edgeconcave by applying interval arithmetic to a simple test of Tardella [100, 132–134]. – aBB underestimators convexify an expression with univariate quadratics [4, 5, 10, 50, 93]; ANTIGONE uses aBB to relax aggregates of bilinear terms [13, 107]. – Term-Specific Underestimators are outlined in Table 1. ANTIGONE organizes the special structure analysis of individual equations using the data structure expression; Figure 2(b) depicts the inheritance structure of this base class.

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Ruth Misener, Christodoulos A. Floudas

Each expression object is a list of terms and associated coefficients; each equation also contains a list of multi-term expression objects forming disjoint vertex sets in the graph representation of the equation. The class RLT equation is derived from equation; its coefficients may change from node to node of the branch-and-cut tree. Objects of type cutting plane may be developed at any node of the branch-and-cut tree; a cutting plane knows the node at which it was created and whether it is globally or locally valid (i.e., whether it may be applied to any node of the branch-and-cut tree or not). For semidefinite programminglike relaxations (see [107]), ANTIGONE also maintains an object with knowledge of every nonlinear term in the MINLP.

3.3 Branch-and-cut Global optimization After the reformulation and special structure detection phases, ANTIGONE initiates a branchand-cut global optimization algorithm that: generates tight convex underestimators; dynamically generates separating hyperplanes; bounds the variables [4, 5, 10, 14, 16, 43, 44, 77, 118, 119, 127, 128]; branches on the search space [2, 16]; finds feasible solutions. This strategy is entirely equivalent to our MISO framework [105]; the additional term types in ANTIGONE do not alter the branch-and-cut implementation because the branch-and-cut phase communicates with the term base class rather than with each individual term type. Refer to the book of Floudas [50] for complete treatment. If ANTIGONE finds, in preprocessing, that an NLP is convex, it will activate a local NLP solver at the root node of the branch-and-cut tree and consider a certificate of optimality from a local solver to be the global optimum. ANTIGONE detects convexity in individual terms and aggregates of quadratic terms; it does not, at this writing, apply function composition rules or symbolic methods (e.g, [56, 111, 145]). For convex problems where the local NLP solver does not return a certificate of optimality at the root node, ANTIGONE reverts to its normal branch-and-cut apprach. In the current software implementation, ANTIGONE interfaces with: CPLEX for LP and MIP relaxations; CONOPT or SNOPT for finding feasible solutions (default: CONOPT); LAPACK [9] for solving linear systems of equations; Boost [22, 23] for validated interval arithmetic. The code is itself built as a library and linked to GAMS via an adaptation of GAMSLinks [88, 144]. 4 Definition of the MINLP Test Suite To validate ANTIGONE, Table 2 defines a test suite of 2571 problems from standard libraries and the open literature; Tables 9 – 18 in the online supplementary material (Appendix A) give more detailed analysis as to the size and complexity of the individual problems. The examples excluded from the standard test libraries are those which either include

ANTIGONE

9

Table 2: MINLP Test Suite of 2571 Problems Problem Class

# Cases

minlp. org

Belgian Chocolate Problem Cascading Tanks Cyclic Scheduling and Control Distillation Sep. Sequences Heat Exchanger Networks Metabolic Networks Multi-Product Batch Plants Periodic Scheduling Pooling Supply Chain Design Three-Echelon Supply Chain Unit Commitment Water Distribution Network Water Treatment Network Crude Oil Scheduling

MIQCQP

BoxQP Circles & Polygon Nesting Crude Oil Scheduling Multi-Period Blending Natural Gas Production Point Packing QAP Randomly-Generated QCQP Reform. GLOBALLib Reform. MINLPLib Standard QP Water Treatment System Water-Using Network

90 35 7 7 3 14 67 135 32 10 15 32 35

AMPL Book Lib Bonmin Test Set GLOBALLib LINLib MacMOOPLib MPLLib MINLPLib Convex PrincetonLib Nonconvex PrincetonLib

34 134 368 39 13 21 249 459 657

Test Libraries

4 8 1 1 3 2 2 13 19 2 8 2 8 18 24

Discrete X X X X X X X X X X X X X X X

X X X

X

Source [38, 39] [60] [48] [28–31] [47, 146] [66, 67] [65, 74, 148] [33, 34] [106] [147, 149] [114, 115, 150] [112, 151] [21, 41] [6, 7, 70, 117] [78, 109, 110] [25, 142, 143] [68] [79, 80] [75, 76] [81, 82, 123] [12] [11, 87] [15] [53, 97, 121] [27, 53] [122] [36, 37] [35, 140, 141]

X

[55] [20, 120] [53, 97]

X

[27, 53]

functional forms that the current ANTIGONE implementation cannot handle (e.g., trigonometric, min/max, errorf) or have known feasible points where the objective value is minus infinity (plus infinity if MINLP is a maximization problem).

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Ruth Misener, Christodoulos A. Floudas

The standard libraries are dynamic entities; we are using the latest versions as of 11 March 2013. The other test cases are from the open literature. The minlp.org test set is unusually difficult; all 115 problems are less than 5 years old and industrially relevant. As of 4 March 2013 there is a group of modeling files labeled Test Set on minlp.org; this is unfortunately unusable as a test set for solver comparisons. Each modeling file may contain multiple MINLP problems and some modeling files require user input arguments. As it is unclear how to use the files labeled Test Set on minlp.org, we have compiled a collection of problems from the minlp.org MINLP Library; Table 2 documents their origin.

5 Testing the Framework To test the software implementation of the ANTIGONE computational framework, we compared the 2571 test problems outlined in Table 2 to several state-of-the-art global optimization solvers available in GAMS 24.2.1: BARON 12.7.3 [138], Couenne 0.4 [16], LINDO 8.0 [58, 85], and SCIP 3.0 [145]. All of the solvers are subject to active development; these are the latest versions as of February 2014. We ran each of the 5 · 2571 = 12855 computational experiments under 4 termination

criteria: (1) a relative optimality gap e = optimality gap e = UB

LB = 1 ⇥ 10

UB LB |LB|

6;

 1 ⇥ 10

6

= 1 ⇥ 10 4 %; (2) an absolute

(3) a 7200 s time limit; (4) an iteration limit of

9 · 109 . No other parameters were altered from default. Following the recommendation of

Dolan and Mor´e [42], each of the 5 solvers addressed a test problem successively so as minimize the effect of fluctuating machine load. We performed our computational studies running 64 bit Linux on an Intel Xeon X5650 2.67 GHz processor with 24 GB RAM. We considered the 2571 test cases outlined in Table 2; the results are in Tables 24 – 43 (online supplement). After solving each of the 2571 test cases using GAMS, a Perl script asserted the feasibility of the solution returned by each solver. Instances where solvers returned infeasible points were automatically relabeled as failures (fl). Therefore, dashes (–) in columns LB and UB of Tables 24 – 43 indicates a benign failure (the solver did not return a solution) while the demarcation fl indicates that the solver returned a point violating a variable bound or constraint equation by max(10 4 , 10 under consideration.

4 · |V |) or more where V

is the value

Comparisons between the solvers are based on performance profiles [42]: t p, s ⌘ Performance metric (e.g., time in seconds) for problem p by solver s 2 S r p, s ⌘ rs (t) =

t p, s ; s2S min t p, s0 : s0 2 S 8 p 2 P 1 size p 2 P : r p, s  t np

ANTIGONE

11

Figures 3 – 16, which diagram rs (log10 (t)) as a function of log10 (t), were generated using the GAMS Performance Tools with options: colselect; useobjest; bnd=1e-4; gaptol=1e-4 (PTOOLS 1.1; www.gamsworld.org/performance/paver/; accessed 11 March 2013). PTOOLS takes trace files returned by the solver and automatically creates performance profiles. The only processing done in transit between trace files generated by the solvers and the PTOOLS input was to penalize solvers for returning infeasible solutions and to add a column for the colselect option. Each of the Figures 3 – 16 has a subfigure (a) and (b). Subfigures (a) are the output of PTOOLS using solver reported time as a resource measurement and the objest option to enforce an objective gap of 10 4 . Paraphrasing the PTOOLS output, a solver is considered optimal if it has the proper model and solver return codes and the relative objective value error is within 10

4

of the best possible solution. If the best found objective value is 0, then

the absolute objective value error is used. Subfigures (b) use the colselect command (and switch off useobjest); this allows us to compare the solvers with respect to their optimality gap at termination (100 · UB|LB|LB ); this column in the GAMS trace file was created

with the same Perl script that asserts feasibility of the solution. Only deterministic lower bounds LB are reported; lower bounds based on guessed variable bounds are discarded.

5.1 Results of the Computational Study Tables 24 – 43 in the online supplement give the results of the computational study with 2571 total problems; Figure 3 displays the performance profiles for the 2112 problems with a nonconvex component (i.e., a second-order test is not enough to prove global optimality). Observe in Figure 3 that the strong performance of ANTIGONE across its entire test suite is the compilation of strong performance across multiple test libraries (Figures 4 – 9). See in Figure 3(b) that ANTIGONE is particularly good at coming to a finite optimality gap for difficult problems; this is useful for contexts where globally optimizing a problem is beyond the reach of current technology but there is still a need to understand bounds on the global solution. Sections 5.2 – 5.7 describe the performance of ANTIGONE with respect to several types of test sets; Sections 5.8 – 5.10 classify the results with respect to types of problems.

5.2 Results: 115 MINLP.ORG Problems Figure 4 and Tables 40 & 41 in the online supplement display results of ANTIGONE on the 115 minlp.org test problems; obeserve that ANTIGONE is unusually strong on these recently-developed, industrially-relevant problems. As an example, ngw-r1-53050 is a large example with: 3471 continuous variables; 1680 discrete variables; 6581 equations; 80 nonlinear terms [114, 115, 150]. While none of the five solvers can globally optimize this

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Ruth Misener, Christodoulos A. Floudas

(a) Time s: Log10 Scale

(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 3: 2112 Total Test Cases with a Nonconvex Component: Performance Profile (a) compares the CPU time for 2112 Total Test Cases with a Nonconvex Component; Profile (b) diagrams the percent gap remaining at 7200 s problem, Table 3 shows that ANTIGONE terminates with a 2.20% gap; ANTIGONE and SCIP find better feasible solutions than the other solvers. Table 3: minlp.org Test Set: Results of ngw-r1-53050 [114, 115, 150]

ANTIGONE 1.1 BARON 12.7.3 Couenne 0.4 LINDO 8.0 SCIP 3.0 Gap ⌘ 100 ·

⇣

UB LB |LB|

Time (s)

Gap

LB

UB

– – – – –

2.20e+00 3.57e+01 4.96e+01 6.20e+01 1.55e+01

3.646e+06 3.684e+06 2.742e+06 2.677e+06 3.151e+06

3.728e+06 5.733e+06 5.437e+06 7.039e+06 3.729e+06

4%

or Time = 7200 s

⌘ ; Termination Criteria: Gap = 1 ⇥ 10

5.3 Results: 482 MIQCQP Problems ANTIGONE performs especially well on the 482 MIQCQP problems from the open literature; see the results in Figure 5 and Tables 42 & 43 in the online supplement. The good performance on the MIQCQP test set is a result of ANTIGONE performing equivalently to GloMIQO in the case of MIQCQP. Table 4 compares solver performance on castro etal 2007 wts Ex16 M1 [36, 37]; this example has 88 continuous variables, 54 equations, and 192 bilinear terms. ANTIGONE is the only solver able to globally opti-

ANTIGONE

13

(a) Time s: Log10 Scale

(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 4: 115 MINLP.ORG Problems: Performance Profile (a) compares the CPU time for 115 MINLP.ORG Problems; Profile (b) diagrams the percent gap remaining at 7200 s mize the problem within the time limit. Couenne returns an infeasible solution and is therefore excluded from Table 4. Table 4: MIQCQP Test Set: Results of castro etal 2007 wts Ex16 M1 [36, 37]

ANTIGONE 1.1 BARON 12.7.3 LINDO 8.0 SCIP 3.0 Gap ⌘ 100 ·

⇣

UB LB |LB|

Time (s)

Gap

LB

UB

26 – – –

1.00e-04 1.85e+01 2.19e+01 1.76e+00

1.359e+03 1.108e+03 1.063e+03 1.335e+03

1.359e+03 1.359e+03 1.361e+03 1.359e+03

4%

or Time = 7200 s

⌘ ; Termination Criteria: Gap = 1 ⇥ 10

5.4 Results: 368 GLOBALLib Problems ANTIGONE performs competitively on the GLOBALLib test set; Figure 6 and Tables 28 & 29 in the online supplement present complete results. For example, Table 5 compares solver performance on a phase equilibrium problem ex6 2 9 [53, 96, 97]. The objective function of ex6 2 9 is (in part): ✓

x0 (31.483 x0 + 6 x1 ) log(4.827 x0 + 0.92 x1 ) + 11.503 x0 log 4.827 x0 + 0.92 x1

◆

.

(2)

ANTIGONE flattens the expression tree of Equation 2 by distributing the first logarithm across the linear expression 31.483 x0 + 6 x1 (see [105]) and reformulating the second loga-

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Ruth Misener, Christodoulos A. Floudas

(a) Time s: Log10 Scale

(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 5: 482 MIQCQP Problems: Performance Profile (a) compares the CPU time for 482 MIQCQP Problems; Profile (b) diagrams the percent gap remaining at 7200 s

rithm via Equation 1: 31.483 · x0 · log x2 + 6 · x1 · log x2 + 11.503 · x0 · log x0

11.503 · x0 · log x2

(3)

where x2 = 4.827 · x0 + 0.92 · x1 is an auxiliary variable. Equation 3 subsequently simplifies: 19.98 · x0 · log x2 + 6 · x1 · log x2 + 11.503 · x0 · log x0

(4)

ANTIGONE replaces the term log x2 in Equation 4 with another auxiliary variable x3 so that this portion of the ex6 2 9 objective function has: two bilinear terms (x0 x3 ; x1 x3 ); one logarithmic term (log x2 ); one univariate composite logarithmic term that is convex across ⇥ ⇤ its entire domain (x0 · log x0 ; x0 2 10 7 , 0.5 ).

Using these automatic reformulations, ANTIGONE globally optimizes ex6 2 9 in 19

seconds. If ANTIGONE was implemented differently, it could detect an additional composite term in Equation 2 because 31.483·x0 +6·x1 is a constant multiple of 4.827·x0 +0.92·x1 . But see in Table 5 that the existing reformulation strategy is very strong with respect to the other solvers. BARON 12.7.3 mistakes ex6 2 9 for an unbounded problem; Couenne 0.4 is excluded from Table 5 because it incorrectly evaluates the objective function; LINDO 8.0 is more than 100 times slower than ANTIGONE; SCIP 3.0 has a 8% gap at the time limit.

5.5 Results: 249 MINLPLib Problems ANTIGONE performs competitively on the MINLPLib test set; the complete results are in Figure 7 and Tables 34 & 35 in the online supplement. Table 6 shows the strong performance

ANTIGONE

15

Table 5: GLOBALLib: Results of ex6 2 9 [53, 96, 97]

ANTIGONE 1.1 BARON 12.7.3 LINDO 8.0 SCIP 3.0 Gap ⌘ 100 ·

⇣

UB LB |LB|

Time (s)

Gap

LB

UB

22 74 3080 –

1.00e-04 – 1.00e-04 7.96e+00

-3.407e-02 • -3.408e-02 -3.678e-02

-3.407e-02 -3.407e-02 -3.407e-02 -3.407e-02

4%

or Time = 7200 s

⌘ ; Termination Criteria: Gap = 1 ⇥ 10

(a) Time s: Log10 Scale

(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 6: 368 GLOBALLib Problems: Performance Profile (a) compares the CPU time for 368 GLOBALLib Problems; Profile (b) diagrams the percent gap remaining at 7200 s

of ANTIGONE on tls4; this example has 17 continuous variables, 89 discrete variables, 65 equations, and 32 nonlinear terms.

Table 6: MINLPLib: Results of tls4 [27]

ANTIGONE 1.1 BARON 12.7.3 Couenne 0.4 LINDO 8.0 SCIP 3.0 Gap ⌘ 100 ·

⇣

UB LB |LB|

Time (s)

Gap

LB

UB

15 473 – – 42

1.00e-04 1.00e-04 5.87e+01 8.31e+01 1.00e-04

8.300e+00 8.300e+00 3.424e+00 1.525e+00 8.300e+00

8.300e+00 8.300e+00 8.300e+00 9.000e+00 8.300e+00

4%

or Time = 7200 s

⌘ ; Termination Criteria: Gap = 1 ⇥ 10

16

Ruth Misener, Christodoulos A. Floudas

(a) Time s: Log10 Scale

(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 7: 249 MINLPLib Problems: Performance Profile (a) compares the CPU time for 249 MINLPLib Problems; Profile (b) diagrams the percent gap remaining at 7200 s

5.6 Results: 1116 PrincetonLib Problems

ANTIGONE performs competitively on the PrincetonLib test set; Figure 8 and Tables 38 & 39 in the online supplement provide complete results. Consider weapon from PrincetonLib; weapon is obviously convex after repeated application of the Equation 1 flatten exponentials reformulation. Table 1 shows that ANTIGONE recognizes convexity and solves weapon quickly. Observe that convexity recognition is a key aspect of the PrincetonLib test set; 459 of the 1116 problems are convex. Figures 8(c) – 8(f) divide the PrincetonLib problems into convex and nonconvex test cases.

Table 7: PrincetonLib: Results of weapon

ANTIGONE 1.1 BARON 12.7.3 Couenne 0.4 LINDO 8.0 SCIP 3.0 Gap ⌘ 100 ·

⇣

UB LB |LB|

Time (s)

Gap

LB

UB

0 0 – – –

1.00e-04 1.00e-04 1.12e+00 1.12e+00 1.12e+00

-1.736e+03 -1.736e+03 -1.755e+03 -1.755e+03 -1.755e+03

-1.736e+03 -1.736e+03 -1.736e+03 -1.736e+03 -1.736e+03

⌘ ; Termination Criteria: Gap = 1 ⇥ 10

4%

or Time = 7200 s

ANTIGONE

17

5.7 Results: 241 Other Library Problems ANTIGONE also performs competitively on a variety of other test libraries (AMPL Book Library; Bonmin; LINLib; MacMOOPLib; MPLLib); See Figure 8 and Tables: 24; 25; 26; 27; 30; 31; 32; 33; 36; 37 (online supplement). Table 8 gives the performance of ANTIGONE on SLay10H; this MIQCQP has 831 continuous variables, 180 discrete variables, 1306 equations, and 20 nonlinear terms. Table 8: Bonmin Test Set: Results of SLay10H [20, 120]

ANTIGONE 1.1 BARON 12.7.3 Couenne 0.4 LINDO 8.0 SCIP 3.0 Gap ⌘ 100 ·

⇣

UB LB |LB|

Time (s)

Gap

LB

UB

52 – – – 3476

1.00e-04 7.70e+01 1.56e+01 6.39e+01 1.00e-04

1.296e+05 1.193e+05 1.200e+05 1.191e+05 1.296e+05

1.296e+05 5.193e+05 1.422e+05 3.301e+05 1.296e+05

4%

or Time = 7200 s

⌘ ; Termination Criteria: Gap = 1 ⇥ 10

5.8 Results: Classes of Nonlinearities Beyond individual libraries, which may contain a wide range of different problem types, it is interesting to consider similar examples across all problem classes. Figures 10 – 12 compare three classes of nonlinearities: quadratic, signomial, and other test cases. Figure 10 represents the performance of the solvers on the 1397 of the 2571 total test cases with exclusively quadratic nonlinearities; Figure 11 diagrams the 759 of the 2571 total test cases that have only signomial nonlinearities but are not in Figure 10; Figure 12 diagrams the remaining instances. From these three figures, it is evident that the strongest component of ANTIGONE is in addressing problems with quadratic nonlinearities. ANTIGONE is equivalent to the MIQCQP solver GloMIQO [102, 104, 107] in the case where all nonlinearities in MINLP are quadratic; ANTIGONE derives the core of its strength from quadratic instances.

5.9 Results: Discrete and Continuous Problems Figure 13 diagrams the 528 incorporating at least one discrete variable; Figure 14 represents solver performance on the 2048 instances without a discrete variable. Observe that ANTIGONE is strong in both categories.

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5.10 Results: Convex and Nonconvex Nonlinearities Figure 15 diagrams the 1781 test cases with nonconvex nonlinearities; Figure 16 represents performance on the 790 test cases on convex NLP and MINLP with nonlinearities that can be proven to be exclusively convex.

6 Conclusion This manuscript introduces ANTIGONE, a general mixed-integer nonlinear global optimization framework. Analysis on a test set of 2571 shows that ANTIGONE performs strongly across an array of standard libraries and problems from the open literature. The performance of ANTIGONE is especially strong with respect to the industrially-relevant minlp.org test set and on MIQCQP. The ANTIGONE framework is the natural evolution of the MIQCQP solver GloMIQO and the MISO framework to general nonconvex MINLP; its extensible framework permits easy integration of new algorithmic advances from the open literature.

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117. J. P. Ruiz and I. E. Grossmann. Water treatment network design, 2009. Available from CyberInfrastructure for MINLP [www.minlp.org, a collaboration of Carnegie Mellon University and IBM Research] at: www.minlp.org/library/problem/index.php?i=24. 118. H. S. Ryoo and N. V. Sahinidis. Global optimization of nonconvex NLPs and MINLPs with applications in process design. Comput. Chem. Eng., 19(5):551–566, 1995. 119. H. S. Ryoo and N. V. Sahinidis. A branch-and-reduce approach to global optimization. J. Glob. Optim., 8:107–138, 1996. 120. N. W. Sawaya. Reformulations, relaxations and cutting planes for generalized disjunctive programming. PhD in Chemical Engineering, Carnegie Mellon University, 2006. 121. A. Saxena, P. Bonami, and J. Lee. Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program., 124(12):383–411, 2010. 122. A. Scozzari and F. Tardella. A clique algorithm for standard quadratic programming. Discrete Applied Mathematics, 156(13):2439 – 2448, 2008. 123. A. Selot, L. K. Kuok, M. Robinson, T. L. Mason, and P. I. Barton. A short-term operational planning model for natural gas production systems. AIChE J., 54(2):495– 515, 2008. 124. H. Sherali, E. Dalkiran, and L. Liberti. Reduced RLT representations for nonconvex polynomial programming problems. J. Glob. Optim., 52(3):447–469, 2012. 125. H. D. Sherali and W. P. Adams. A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, Dordrecht, Netherlands, 1999. 126. H. D. Sherali and A. Alameddine. A new reformulation-linearization technique for bilinear programming problems. J. Glob. Optim., 2:379 – 410, 1992. 127. H. D. Sherali and C. H. Tuncbilek. A reformulation-convexification approach for solving nonconvex quadratic-programming problems. J. Glob. Optim., 1995(7):1, 131. 128. H. D. Sherali and C. H. Tuncbilek. New reformulation linearization/convexification relaxations for univariate and multivariate polynomial programming problems. Oper. Res. Letters, 21(1):1 – 9, 1997. 129. V. Shikhman and O. Stein. On jet-convex functions and their tensor products. Optim., 61(6):717–731, 2012. 130. E. M. B. Smith and C. C. Pantelides. A symbolic reformulation/spatial branch-andbound algorithm for the global optimisation of nonconvex MINLPs. Comput. Chem. Eng., 23(4 - 5):457 – 478, 1999. 131. B. Tadayon and J. C. Smith. Algorithms for an integer multicommodity network flow problem with node reliability considerations. J. Optim. Theory Appl., 2013. doi: 10.1007/s10957-013-0378-5. URL http://dx.doi.org/10.1007/ s10957-013-0378-5. 132. F. Tardella. On a class of functions attaining their maximum at the vertices of a polyhedron. Discret. Appl. Math., 22:191–195, 1988/89. 133. F. Tardella. On the existence of polyhedral convex envelopes. In C. A. Floudas and P. M. Pardalos, editors, Frontiers in Global Optimization, pages 563–573. Kluwer Academic Publishers, 2003. 134. F. Tardella. Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett., 2:363–375, 2008.

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135. M. Tawarmalani and N. V. Sahinidis. Semidefinite relaxations of fractional programs via novel convexification techniques. J. Glob. Optim., 20:133–154, 2001. 136. M. Tawarmalani and N. V. Sahinidis. Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Applications, Software, and Applications. Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, Norwell, MA, USA, 2002. 137. M. Tawarmalani and N. V. Sahinidis. Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. Math. Program., 99:563–591, 2004. 138. M. Tawarmalani and N. V. Sahinidis. A polyhedral branch-and-cut approach to global optimization. Math. Program., 103:225–249, 2005. 139. M. Tawarmalani, J.-P. P. Richard, and C. Xiong. Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program., 138(1-2):531–577, 2013. 140. J. Teles, P. M. Castro, and A. Q. Novais. Lp-based solution strategies for the optimal design of industrial water networks with multiple contaminants. Chem. Eng. Sci., 63 (2):376 – 394, 2008. 141. J. P. Teles, P. M. Castro, and H. A. Matos. Global optimization of water networks design using multiparametric disaggregation. Comput. Chem. Eng., 40(0):132 – 147, 2012. 142. D. Vandenbussche and G. L. Nemhauser. A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program., 102(3):559–575, 2005. 143. D. Vandenbussche and G. L. Nemhauser. A polyhedral study of nonconvex quadratic programs with box constraints. Math. Program., 102(3):531–557, 2005. 144. S. Vigerske. COIN-OR/GAMSLinks. https://projects.coin-or.org/ GAMSlinks/, 2011. 145. S. Vigerske. Decomposition in Multistage Stochastic Programming and a Constraint Integer Programming Approach to Mixed-Integer Nonlinear Programming. PhD in Mathematics, Humboldt-University Berlin, 2012. 146. T. F. Yee and I. E. Grossmann. Simultaneous optimization models for heat integrationII. Heat exchanger network synthesis. Comput. Chem. Eng., 14(10):1165 – 1184, 1990. 147. F. You and I. E. Grossmann. Mixed-integer nonlinear programming models and algorithms for large-scale supply chain design with stochastic inventory management. Ind. Eng. Chem. Res., 47(20):7802–7817, 2008. 148. F. You and I. E. Grossmann. Mixed-integer nonlinear programming models for the optimal design of multi-product batch plant, 2009. Available from CyberInfrastructure for MINLP [A collaboration of CMU and IBM Research] at: www.minlp.org/library/problem/index.php?i=48. 149. F. You and I. E. Grossmann. Mixed-integer nonlinear programming models and algorithms for supply chain design with stochastic inventory management, 2009. Available from CyberInfrastructure for MINLP [A collaboration of CMU and IBM Research] at: www.minlp.org/library/problem/index.php?i=30. 150. F. You and I. E. Grossmann. Integrated multi-echelon supply chain design with inventories under uncertainty: Minlp models, computational strategies. AIChE Journal, 56 (2):419–440, 2010. 151. E. Zondervan and I. E. Grossmann. A deterministic security constrained unit commitment model, 2009. Available from CyberInfrastructure for MINLP [A collaboration of CMU and IBM Research] at: www.minlp.org/library/problem/index.php?i=41.

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(a) 1116 Cases; Time s

(b) 1116 Cases; Remaining Gap at 7200 s

(c) 459 Convex Cases; Time s

(d) 459 Convex Cases; Remaining Gap at 7200 s

(e) 657 Nonconvex Cases; Time s

(f) 657 Nonconvex Cases; Remaining Gap at 7200 s

Fig. 8: 1116 PrincetonLib Problems: Performance Profiles (a), (c), (e) compare the CPU time for 1116 PrincetonLib Problems; Profiles (b), (d), (f) diagram the percent gap remaining at 7200 s. All graphs are on a Log10 Scale.

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(a) Time s: Log10 Scale

(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 9: 241 Other Library Problems: Performance Profile (a) compares the CPU time for 241 Other Library Problems; Profile (b) diagrams the percent gap remaining at 7200 s

(a) Time s: Log10 Scale

(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 10: 1397 Problems with Exclusively Quadratic Nonlinearities: Performance Profile (a) compares the CPU time for the 1397 test cases with exclusively quadratic nonlinearities; Profile (b) diagrams the percent gap remaining at 7200 s

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(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 11: 759 Problems with a Signomial Component: Performance Profile (a) compares the CPU time for the 759 test cases with a signomial component; Profile (b) diagrams the percent gap remaining at 7200 s

(a) Time s: Log10 Scale

(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 12: 415 Problems with Exponential, Logarithmic, or Power Terms: Performance Profile (a) compares the CPU time for the 415 test cases with an exponential, logarithmic, or power terms; Profile (b) diagrams the percent gap remaining at 7200 s

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(a) Time s: Log10 Scale

(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 13: 528 Problems with a Discrete Component: Performance Profile (a) compares the CPU time for the 528 test cases with a discrete component; Profile (b) diagrams the percent gap remaining at 7200 s

(a) Time s: Log10 Scale

(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 14: 2048 Problems with No Discrete Component: Performance Profile (a) compares the CPU time for the 2048 test cases with no discrete component; Profile (b) diagrams the percent gap remaining at 7200 s

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(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 15: 1781 Problems with Nonconvex Nonlinearities: Performance Profile (a) compares the CPU time for the 1781 test cases with a nonconvex component; Profile (b) diagrams the percent gap remaining at 7200 s

(a) Time s: Log10 Scale

(b) Remaining Gap at 7200 s: Log10 Scale

Fig. 16: 790 Convex NLP and MINLP with Exclusively Convex Nonlinearities: Performance Profile (a) compares the CPU time for the 790 test cases with all convex nonlinearities; Profile (b) diagrams the percent gap remaining at 7200 s