## Reformulation and Convex Relaxation Techniques for Global Optimization (2004)

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Venue: | 4OR |

Citations: | 9 - 7 self |

### BibTeX

@TECHREPORT{Liberti04reformulationand,

author = {Leo Sergio Liberti},

title = {Reformulation and Convex Relaxation Techniques for Global Optimization},

institution = {4OR},

year = {2004}

}

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### Abstract

Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested in determining the globally optimal point. This thesis is concerned with techniques for establishing such global optima using spatial Branch-and-Bound (sBB) algorithms.

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Citation Context ...oth the objective function�and the feasible regionªare convex. This is an interesting class of problems as it is possible to show that any local solution of a convex problem is also a global solution =-=[85, 98]-=-. As in the linear case, convex reformulations are rarely exact. However, convex relaxations (which will be analysed in more detail in Section 2.3) are used within more complex procedures for the solu... |

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Citation Context ...he 1970s and early 1980s [74, 77]. It is worth noting that the first deterministic global optimization technique that was able to deal with generic nonconvex continuous NLPs was Interval Optimization =-=[48, 49]-=-). Unfortunately, it often has slow convergence due to the fact that Interval Arithmetic generally provides very wide intervals for the objective functionsChapter 1. Introduction 22 value. Toward the ... |

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Citation Context ...seudo-convexity condition that allows the ECP method to derive a new valid cutting plane at each iteration. In the stochastic global optimization field, recent advances include Differential Evolution =-=[119]-=-, Adaptive Lagrange-Multiplier Methods [140], Simulated and Nested Annealing [96], Ant Colony Simulation [78, 25], Quantum Dynamics in complex biological evolution [52], Quantum Thermal Annealing [68]... |

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Citation Context ...roblems are called unconstrained problems, though in fact a truly unconstrained optimization problem would lack variable bounds as well). This is a very well-studied and interesting class of problems =-=[35, 14, 38]-=-, and much effort has gone into reformulations of other types of problems to this type. Moreover, most stochastic optimizationsChapter 2. Overview of reformulation techniques in optimization 33 method... |

281 |
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Citation Context ...roblems to binary problems A discrete optimization problem is such that the problem variables can take only discrete values. It is possible to reformulate discrete problems to binary problems exactly =-=[27, 144, 110]-=-. Let Úbe a discrete problem variables taking values in the set�Ú�����Ú��. By introducing�new binary variables¬���¬�, we can replaceÚbyÈ���Ú�¬�and add a linear constraintÈ���¬�� to the definition of t... |

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Citation Context ...ich provide them with the necessary information; for example, Smith’s [114] code was embedded in the gPROMS modelling tool [30] while the BARON code [102] has recently been made available within GAMS =-=[22]-=-. Whilst these developments are undeniably useful from the immediate practical point of view, the wider dissemination of global optimization technology requires a different approach which allows the s... |

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Citation Context ... the problem is separable. The definition of the feasible regionªvaries. Usually [14, 35, 79],ªis defined by a system of separable inequalitiesÈÒ�����Ü����,���Ñ, where����Ê�Êfor each���. Some authors =-=[27, 58]-=- requireªto be a polytope. 2.1.2.1 Separation of bilinear forms In the procedure of Section 2.1.8.1, the basic idea is the separation of a bilinear formÜÝ. This idea rests on the relationshipÜÝ�Ü ÜÝÝ ... |

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Citation Context ... the stochastic global optimization field, recent advances include Differential Evolution [119], Adaptive Lagrange-Multiplier Methods [140], Simulated and Nested Annealing [96], Ant Colony Simulation =-=[78, 25]-=-, Quantum Dynamics in complex biological evolution [52], Quantum Thermal Annealing [68], Ruin and Recreate Principle [105] and Tabu Search [23]. 1.3.3 Two-phase global optimization algorithms Most met... |

210 |
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Citation Context ...o appear on the subject. They explained the basics of local optimization (Lagrange multipliers and Karush-Kuhn-Tucker conditions) and some of the early techniques in deterministic [92] and stochastic =-=[131]-=- global optimization. Early topics in deterministic global optimization include convex optimization [94], Branch-and-Bound techniques restricted to particular classes of problems (e.g. concave minimiz... |

207 |
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Citation Context ... search phase, under the most general conditions global optimization of NLPs is an NP-hard problem. The only non-iterative method for global optimization, based on Gröbner bases [47], is also NP-hard =-=[26]-=-. This research focuses on deterministic algorithms for global optimization. In particular, the Branch-and-Select family of algorithms promises a good performance in terms of computational 4 I.e., obj... |

190 |
Combinatorial Optimization: Theory and Algorithms
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Citation Context ...re we consider a fast graph-theoretical algorithm for the identification of linear constraint subsetsÄthat satisfy (3.23) for a given multiplier variableÞÐ. We start by constructing a bipartite graph =-=[64, 45, 51]-=-�Ðwhere the set of nodes is partitioned into two disjoint subsetsÆÄandÆÎ Ð. We call these the “constraint” and “variable” nodes respectively. The former correspond to the ÆÎ Ð��������Ð�� �� set of lin... |

144 | The maximum clique problem
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Citation Context ...em as a minimization problem with nonnegativity constraints. Some regularity conditions ensure that a stationary point of the reformulation is a solution of the original problem. Bomze and co-workers =-=[20, 21]-=- reformulated the maximum clique problem as a quadratic problem over a standard simplex, and used copositivity-based procedures to solve the standardized quadratic problem. In [19], a general method f... |

128 |
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Citation Context ...r and Jacobi all worked on this problem. IfØ�, then the form is called positive definite; then the form is called semidefinite (these terms were introduced by Gauss in his Disquisitiones arithmeticae =-=[62]-=-). Their work is also relevant to the theory of Ý ���ÝÖ ÝÖ ��� ÝÖØ (2.5) convex, concave and d.c. functions, since a positive quadratic term is a convex function and a negative quadratic term is a con... |

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Citation Context ...in this period dealt either with applications of global optimization to very specific cases, or with theoretical results concerning convergence proofs. One notable exception was the work of McCormick =-=[80]-=- who considered symbolic transformations of problems: his methods are such that they can, in theory, be carried out automatically by a computer. A major reason for the slow pace of progress in continu... |

98 |
Set Theory
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Citation Context ... to devise an exact reformulation of a problem in ÊÒto a problem inÊ, i.e. having a single problem variable. Given a space-filling curve��Ê���℄Ò(which exists by transfinite cardinality considerations =-=[65, 24]-=-), the dimensionality of the problem can be reduced fromÒto 1 by solving the reduced problemÑ�ÒÝÊ��Ý. The function�Æ�mapsÊintoÊ, and may thus be minimized via efficient 1-dimensional optimization meth... |

96 |
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Citation Context ...roblems to binary problems A discrete optimization problem is such that the problem variables can take only discrete values. It is possible to reformulate discrete problems to binary problems exactly =-=[27, 144, 110]-=-. Let Úbe a discrete problem variables taking values in the set�Ú�����Ú��. By introducing�new binary variables¬���¬�, we can replaceÚbyÈ���Ú�¬�and add a linear constraintÈ���¬�� to the definition of t... |

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Citation Context ...ollowing inequality constraints are inserted in the relaxed problem: Û��ÜÄÝÝÍÜ ÜÄÝÍ� ÜÄÝÄ�ÜÍÝ toÜÍ ÜÄÝÍ ÝÄ � The above linear inequalities have been shown to be the convex envelope of a bilinear term =-=[10]-=-. The maximum separation of the bilinear termÜÝfrom its convex envelopeÑ�ÜÜÄÝ ÝÄÜsÝÍÜ ÜÍÝÍinside the rectangle�ÜÄ�ÜÍ℄¢�ÝÄ�ÝÍ℄occurs at the middle point ÜÄÜÍ�ÝÄÝÍand is equal [13].sChapter 2. Overview ... |

60 |
The consistent initialization of differential-algebraic systems
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Citation Context ...ination of dilations in bipartite graphs. 3.3.3 Efficient identification of dilations Dilations in bipartite graphs are closely related to the existence of output set assignments (OSA) in such graphs =-=[87]-=-. A subset�Ðof the edges�Ðin graph�Ðis an OSA if no node is incident to more than one edge in�Ð. A complete OSA is one in which each and every node� ÆÄis incident to exactly one edge in it; in this ca... |

56 |
Galois Theory
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Citation Context ...he symmetric group of order 5) which is not soluble since its biggest proper normal subgroup is��, the smallest non-soluble group. For details on Galois theory and the solvability of polynomials, see =-=[118]-=-. 4.3 The roots ofÉ�Üand in�stheir uniqueness UnlikeÈ�Ü��, the polynomialÉ�Üdoes not depend on the range ofÜbeing considered. Moreover, as shown formally in Section 4.3.1 below,É�Ühas exactly one real... |

51 |
Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables
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Citation Context ...s offers a very fast way to calculate lower bounds on the objective function: it is no surprise that the first Branch-and-Bound approaches to global optimization were restricted to separable problems =-=[34, 117, 15]-=-. ª. A function��ÊÒ�Êis separable if and only if there are �������Ò�Ê�Êsuch that for allÜªwe have�Ü�ÈÒ����Ü�. If the objective function�of a problem (2.1) is separable, then the problem is separable. ... |

51 |
On algorithms for obtaining a maximum transversal
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Citation Context ...y identifying an unassigned node� ÆÄ(i.e. one that is not incident to that�¬��¬��Ð��¬�����«and�¬��¬s�����������������«��« ��«��« any edge in�Ð) and tracing an augmenting path emanating from this node =-=[29]-=-. An augmenting path is a sequence of« edges of the form: such �Ð��¬�����«. If such a path can be found, then an OSA of cardinalityÑ can be obtained from�Ðsimply by replacing the«sedges�¬��¬ �¬�����«i... |

51 |
Global optimization of nonconvex NLPs and MINLPs with applications in process design
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Citation Context ... The first method that was able to deal directly with the generic nonconvex NLPs in the form (1.1) was Ryoo and Sahinidis’ Branch-and-Reduce algorithm which appeared in an paper published in May 1995 =-=[99, 100]-=-. Shortly afterward, Floudas’ team published their first article on the«BB branch-and-bound method [13] which was then thoroughly explored and analysed in several subsequent papers by Adjiman [7, 6, 3... |

49 |
A branch-and-reduce approach to global optimization
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(Show Context)
Citation Context ... The first method that was able to deal directly with the generic nonconvex NLPs in the form (1.1) was Ryoo and Sahinidis’ Branch-and-Reduce algorithm which appeared in an paper published in May 1995 =-=[99, 100]-=-. Shortly afterward, Floudas’ team published their first article on the«BB branch-and-bound method [13] which was then thoroughly explored and analysed in several subsequent papers by Adjiman [7, 6, 3... |

47 |
A.: A new reformulation-linearization technique for bilinear programming problems
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(Show Context)
Citation Context ...might not be the best possible. 2.4.1 Reformulation-linearization technique The basic idea of the Reformulation-Linearization Technique (RLT), proposed by Sherali and co-workers in a number of papers =-=[109, 111, 107, 113, 112, 110, 108]-=-, consists in deriving valid cuts to the problem by multiplying together various factors involving variables and constraints. This technique was initially Ñ�ÒÜÜÌÉÜ proposed in conjunction with combina... |

45 | Deterministic Global Optimization
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- 2000
(Show Context)
Citation Context ...ofÀ�Ü. The most promising method to this end seems to be Interval Matrix Analysis. VariousÓÒandÓÒmethods have been proposed to solve both the uniform and the non-uniform diagonal shift matrix problem =-=[39]-=-. Thus, having constructed a convex underestimating function for the reformulated function �Ü, the relaxation of the problem is carried out accordingly, bearing in mind that:sChapter 2. Overview of re... |

43 |
Convex Analysis and Global Optimization
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- 1998
(Show Context)
Citation Context ...man Problem [9] but were very soon applied to continuous and mixed-integer nonlinear optimization [34, 59, 92]. In this section, we present a general theory of such algorithms (based on material from =-=[133]-=-), together with the necessary convergence proofs. Branch-and-Select algorithms can be used to solve the widest possible class of optimization problems (1.1) to global optimality, even when objective ... |

41 |
Constrained Global Optimization: Algorithms and Applications
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(Show Context)
Citation Context ...ry textbooks began to appear on the subject. They explained the basics of local optimization (Lagrange multipliers and Karush-Kuhn-Tucker conditions) and some of the early techniques in deterministic =-=[92]-=- and stochastic [131] global optimization. Early topics in deterministic global optimization include convex optimization [94], Branch-and-Bound techniques restricted to particular classes of problems ... |

41 |
A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs
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- 1999
(Show Context)
Citation Context ...B algorithm, a number of Branch-and-Select algorithms geared toward the most generic nonconvex MINLP formulation appeared in the literature, like Smith and Pantelides’ symbolic reformulation approach =-=[114, 115, 116]-=-, Pistikopoulos’ Reduced Space Branch-and-Bound approach [32] (which only applies to continuous NLPs), Grossmann’s Branch-and-Contract algorithm [146] (which also only applies to continuous NLPs) and ... |

39 |
An algorithm for separable nonconvex programming problems
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(Show Context)
Citation Context ...problems were devoted to discrete problems such as the Travelling Salesman Problem (TSP). The first paper concerning continuous global optimization with a BB (deterministic) technique dates from 1969 =-=[34]-=-. In the 1970s and 1980s, work on continuous or mixed-integer deterministic global optimization was scarce. Most of the papers published in this period dealt either with applications of global optimiz... |

36 | Floudas, Rigorous convex underestimators for general twicedifferentiable problems
- Adjiman, A
- 1996
(Show Context)
Citation Context ...99, 100]. Shortly afterward, Floudas’ team published their first article on the«BB branch-and-bound method [13] which was then thoroughly explored and analysed in several subsequent papers by Adjiman =-=[7, 6, 3, 4, 5, 8]-=-. The first«BB variant that addressed problems in the form (1.1) appeared in 1997 [3]. One notable limitation of the«BB algorithm is that it relies on the functions being twice differentiable in the c... |

34 | Heuristic algorithms for the unconstrained binary quadratic programming problem. Working Paper, The Management School
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- 1998
(Show Context)
Citation Context ...inary problems Although this reformulation only applies to a very special class of optimization problems, it is one of the very few exact linear1 Ñ�Ü ×�Ø� �����ÒÝ���Ü� ÕÌÝ reformulations �����ÒÝ���Ü� =-=[16, 83]-=-. Any unconstrained quadratic binary problemÑ�ÜÜ���ÒÜÌÉÜcan be reformulated �����ÒÝ���Ü�Ü�sexactly to: Ü���Ò�Ý���Ò� 1 Since the considered problems are binary, calling this a linear relaxation is a sl... |

34 |
Global optimization of mixed integer nonlinear programs: A theoretical and computational study
- Tawarmalani, Sahinidis
(Show Context)
Citation Context ...2.1.8 Factorable problems Most common functions can be expressed in factorable form [80]; this form is desirable because it makes it relatively easy to construct convex relaxations in a recursive way =-=[126, 141]-=-. 2 If there are any cusp points, the function might fail to alternate in such a way; in this case we simply define an empty interval where�����.sÑ�ÒÜÊÒ �ÆÜ Chapter 2. Overview of reformulation techni... |

30 | Floudas. Finding all solutions of nonlinearly constrained systems of equations
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(Show Context)
Citation Context ... 51 ÛÌ�ÜÝÍÞÍ For a trilinear termÜÝÞa ÛÌ�ÜÝÄÞÄÜÄÝÞÍ ÜÍÝÞÄÜÍÝÄÞ ÜÍÝÄÞÄ ÜÍÝÍÞÍ new variableÛÌis ÜÄÝÍÞ ÜÄÝÍÞÍ ÜÄÝÄÞÄ introduced, to replace the trilinear termÜÝÞ, together with the following constraints =-=[75]-=-: ÛÌ�ÜÝÍÞÄÜÍÝÞÍ ÛÌ�ÜÝÄÞÍ ÛÌ�ÜÝÄÞÍ ÜÄÝÞÄÜÍÝÄÞ ÜÍÝÄÞÍ ÜÄÝÄÞÄ ÜÄÝÞÍ ÜÍÝÍÞ ÜÄÝÄÞÍ ÜÍÝÍÞÍ ÜÄÝÍÞ ÜÄÝÍÞÄ ÜÍÝÍÞÍ ÛÌ�ÜÝÍÞÄÜÍÝÞÄÜÄÝÄÞ ÜÍÝÍÞÄ ÜÄÝÄÞÄ ÛÌ�ÜÝÍÞÍ ÜÍÝÞÍ ÜÍÝÍÞ ÜÍÝÍÞÍ Û�� ÜÄÝ ÝÍ ÜÄÝ Ü ÝÍ ÜÄ ÝÄÝÍ Ü ÝÍÜÄ... |

28 |
Checking local optimality in constrained quadratic programming is NP-hard
- Pardalos, Schnitger
- 1987
(Show Context)
Citation Context ... local search will find the global optimum. This procedure is iterative in nature so that the search space can be explored exhaustively. Local optimization of nonconvex problems is an NP-hard problem =-=[93]-=-. Because all iterative methods for global optimization rely on a local search phase, under the most general conditions global optimization of NLPs is an NP-hard problem. The only non-iterative method... |

28 | Sahinidis. Semidefinite relaxations of fractional programs via novel techniques for constructing convex envelopes of nonlinear functions - Tawarmalani, V |

26 |
Tabu Search applied to global optimization
- Chelouah, Siarry
- 2000
(Show Context)
Citation Context ...ted and Nested Annealing [96], Ant Colony Simulation [78, 25], Quantum Dynamics in complex biological evolution [52], Quantum Thermal Annealing [68], Ruin and Recreate Principle [105] and Tabu Search =-=[23]-=-. 1.3.3 Two-phase global optimization algorithms Most methods for global optimization are based on two phases [104]: a global search phase and a local search phase. The reason for this is that there a... |

22 |
On standard quadratic optimization problems
- Bomze
- 1998
(Show Context)
Citation Context ...em as a minimization problem with nonnegativity constraints. Some regularity conditions ensure that a stationary point of the reformulation is a solution of the original problem. Bomze and co-workers =-=[20, 21]-=- reformulated the maximum clique problem as a quadratic problem over a standard simplex, and used copositivity-based procedures to solve the standardized quadratic problem. In [19], a general method f... |

22 | Global Optimization in Engineering Design - Grossmann, editor - 1996 |

21 |
Optimization by direct search and systematic reduction of the size of the search region, AIChE
- Luss, Jaakola
- 1973
(Show Context)
Citation Context ....g. concave minimization problems [53]) and some theoretical and complexity-related studies [135, 97]. Stochastic algorithms based on adaptive random search appeared between the 1970s and early 1980s =-=[74, 77]-=-. It is worth noting that the first deterministic global optimization technique that was able to deal with generic nonconvex continuous NLPs was Interval Optimization [48, 49]). Unfortunately, it ofte... |

21 |
breaking optimization results using the ruin and recreate principle
- Schrimpf, Schneider, et al.
(Show Context)
Citation Context ... Methods [140], Simulated and Nested Annealing [96], Ant Colony Simulation [78, 25], Quantum Dynamics in complex biological evolution [52], Quantum Thermal Annealing [68], Ruin and Recreate Principle =-=[105]-=- and Tabu Search [23]. 1.3.3 Two-phase global optimization algorithms Most methods for global optimization are based on two phases [104]: a global search phase and a local search phase. The reason for... |

21 | New properties and computational improvement of the GOP algorithm for problems with quadratic objective function and constraints
- Floudas, Visweswaran
- 1993
(Show Context)
Citation Context ...he problems of Table 3.1 using eight different codes. The first six are codes described in the literature:sChapter 3. Reduction constraints for sparse bilinear programs 84 1 = [40] (Foulds, 1992) 2 = =-=[138]-=- (Visweswaran, 1993) 3 = [17] (Ben-Tal, 1994) 4 = [139] (Visweswaran, 1996) 5 = [1] (Adhya, 1999) 6 = [126] (Tawarmalani, 1999) and their performance is shown in Table 3.2, taken from the correspondin... |

20 |
Dynamic Tunneling Algorithm for Global Optimization
- Yao
- 1989
(Show Context)
Citation Context ... there were also theoretical studies [50, 55] and parallel implementation studies and reports [90, 42, 33]. In the area of stochastic optimization, some of the first algorithms employing “tunnelling” =-=[145]-=- began to appear in the same period. Since the beginning of the 1990s, the optimization research community has witnessed an explosion of papers, books, algorithms, software packages and resources conc... |