A compact variant of the QCR method for quadratically constrained quadratic 0-1 programs

Quadratic Convex Reformulation (QCR) is a technique that was originally proposed for quadratic 0-1 programs, and then extended to various other problems. It is used to convert non-convex instances into convex ones, in such a way that the bound obtained by solving the continuous relaxation of the reformulated instance is as strong as possible. In this paper, we focus on the case of quadratically constrained quadratic 0-1 programs. The variant of QCR previously proposed for this case involves the addition of a quadratic number of auxiliary continuous variables. We show that, in fact, at most one additional variable is needed. Some computational results are also presented

A binarisation heuristic for non-convex quadratic programming with box constraints

Non-convex quadratic programming with box constraints is a fundamental problem in the global optimization literature, being one of the simplest NP-hard nonlinear programs. We present a new heuristic for this problem, which enables one to obtain solutions of excellent quality in reasonable computing times. The heuristic consists of four phases: binarisation, convexification, branch-and-Bound, and local optimisation. Some very encouraging computational results are given

On Linearising Mixed-Integer Quadratic Programs via Bit Representation

It is well known that, under certain conditions, one can use bit representation to transform both integer quadratic programs and mixed-integer bilinear programs into mixed-integer linear programs (MILPs), and thereby render them easier to solve using standard software packages. We show how to convert a more general family of mixed-integer quadratic programs to MILPs, and present several families of strong valid linear inequalities that can be used to strengthen the continuous relaxations of the resulting MILPs

A Binarisation Heuristic for Non-Convex Quadratic Programming with Box Constraints

Non-convex quadratic programming with box constraints is a fundamental problem in the global optimization literature, being one of the simplest NP-hard nonlinear programs. We present a new heuristic for this problem, which enables one to obtain solutions of excellent quality in reasonable computing times. The heuristic consists of four phases: binarisation, convexification, branch-and-bound, and local optimisation. Some very encouraging computational results are given

New valid inequalities and facets for the simple plant location problem

The Simple Plant Location Problem is a well-known (and NP-hard) combinatorial optimisation problem, with applications in logistics. We present a new family of valid inequalities for the associated family of polyhedra, and show that it contains an exponentially large number of new facet-defining members. We also present a new procedure, called facility augmentation, which enables one to derive even more valid and facet-defining inequalities

Bit Representation Can Improve SDP Relaxations of Mixed-Integer Quadratic Programs

A standard trick in integer programming is to replace bounded integer variables with binary variables, using a bit representation. In a previous paper, we showed that this process can be used to improve linear programming relaxations of mixed-integer quadratic programs. In this paper, we show that it can also be used to improve {\em semidefinite}\/ programming relaxations

Urban Aboriginal identity construction in Australia: an Aboriginal perspective utilising multi-method qualitative analysis

ABSTRACT Background: Since British arrival, Aboriginal people have experienced marginalisation and extreme disadvantage within Australian society. Urban-based Aboriginal people, even more than those living in remote communities, have been subject to the impact of racism and discrimination on self-identity. Nonetheless, many urban-based Aboriginal people proudly identify with their Aboriginality. Having long been the subject of others’ research, it is only in recent times that the question of identity has attracted attention in Aboriginal research. Furthermore, few studies have addressed urban Aboriginality from an insider’s perspective. Aim and significance: The main aim of this research was to understand better the process of the construction of Aboriginal identity. Knowing how Aboriginal people see themselves and their future as Aboriginal within the broader Australian community is significant in providing a foundation for both the protection and the preservation of urban-based Aboriginal identity, while helping to create positive practical benefits and minimising the damage to Aboriginal culture that result from collective memory loss. A secondary aim was to test whether tools of narrative analysis could be used within an Indigenous Australian context, utilising Aboriginal Australian English language, and in the context of a specific urban setting. Method: The study used purposeful sampling to recruit 11 individuals from three age cohorts of mixed-descent Aboriginal people living in urban communities on the south coast of New South Wales, Australia. Data were collected through in-depth semi-structured interviews which were tape-recorded and then transcribed in full. Both thematic and narrative methods were employed to analyse the data. Interpretations benefitted from an insider perspective, as the researcher is a member of the community under study. Results: Findings from both methods of analysis show that participants experience their Aboriginality as problematic. Nonetheless, they make strong claims to Aboriginal identity. In making such claims, they link the personal to the social in a variety of ways, drawing on both negative and positive aspects of being part of a marginalised culture to explain the construction of the problem of Aboriginal identity and, as importantly, its on-going resolution through processes of identity construction and re-construction. The Shoalhaven Aboriginal worldview is revealed thorough a thematic analysis of 11 interviews and shows that participants are able to construct positive versions of self when they perceive themselves as living in accordance with the prescribed worldview. Results from case study analyses reveal how four participants distinctly craft the Shoalhaven worldview. The adoption of multi-method qualitative analysis documents the construction of both collective and personal Aboriginal identities and shows how these become core elements of the various strategies for solving the broader problems of Aboriginal identity in contemporary urban Australian society. Conclusion: Understanding the construction of Aboriginal identity from a micro-sociological perspective, with the added benefit of an insider’s analysis, can point the way to the development of more meaningful and appropriate strategies to both address and alleviate the broader problems of Aboriginal marginalisation in Australia. The findings from this research have documented the narrative construction of urban Aboriginal identity revealing the positive and negative aspects of the urban Aboriginal identity concept. A starting point to address the broader problem of Aboriginal marginalisation in Australia is to focus on the positive elements of the urban Aboriginal identity concept, with a view to devise, develop and implement culturally appropriate strategies and policies. The researcher’s life experience, informed by the ontology (collective values and perspectives) of the community, influenced and informed the analysis and results of the study. This shared ontology and community acceptance was integral in the process of developing and maintaining rapport and trust with participants which ultimately shaped the interaction process influencing personal accounts told in the interview

On the Lovász theta function and some variants

The Lovász theta function of a graph is a well-known upper bound on the stability number. It can be computed efficiently by solving a semidefinite program (SDP). Actually, one can solve either of two SDPs, one due to Lovász and the other to Gr�ötschel et al. The former SDP is often thought to be preferable computationally, since it has fewer variables and constraints. We derive some new results on these two equivalent SDPs. The surprising result is that, if we weaken the SDPs by aggregating constraints, or strengthen them by adding cutting planes, the equivalence breaks down. In particular, the Gr�ötschel et al. scheme typically yields a stronger bound than the Lovász one

On the Lovász theta function and some variants

The Lovász theta function of a graph is a well-known upper bound on the stability number. It can be computed efficiently by solving a semidefinite program (SDP). Actually, one can solve either of two SDPs, one due to Lovász and the other to Gr�ötschel et al. The former SDP is often thought to be preferable computationally, since it has fewer variables and constraints. We derive some new results on these two equivalent SDPs. The surprising result is that, if we weaken the SDPs by aggregating constraints, or strengthen them by adding cutting planes, the equivalence breaks down. In particular, the Gr�ötschel et al. scheme typically yields a stronger bound than the Lovász one

On Linearising Mixed-Integer Quadratic Programs via Bit Representation

It is well known that, under certain conditions, one can use bit representation to transform both integer quadratic programs and mixed-integer bilinear programs into mixed-integer linear programs (MILPs), and thereby render them easier to solve using standard software packages. We show how to convert a more general family of mixed-integer quadratic programs to MILPs, and present several families of strong valid linear inequalities that can be used to strengthen the continuous relaxations of the resulting MILPs