Copositive optimization - Recent developments and applications

Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of optimization is impressive, since problem classes both in the continuous and discrete world, as well as both deterministic and stochastic models are covered. Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NP-hard combinatorial optimization problems. Here some of the recent success stories are told, along with principles, algorithms and applications.     

Selection of a representative value function in robust multiple criteria ranking and choice

We introduce the concept of a representative value function in robust ordinal regression applied to multiple criteria ranking and choice problems. The proposed method can be seen as a new interactive UTA-like procedure, which extends the UTA^GMS and GRIP methods. The preference information supplied by the decision maker (DM) is composed of a partial preorder and intensities of preference on a subset of reference alternatives. Robust ordinal regression builds a set of general additive value functions which are compatible with the preference information, and returns two binary preference relations: necessary and possible. They identify recommendations which are compatible with all or at least one compatible value function, respectively. In this paper, we propose a general framework for selection of a representative value function from among the set of compatibles ones. There are a few targets which build on results of robust ordinal regression, and could be attained by a representative value function. In general, according to the interactively elicited preferences of the DM, the representative value function may emphasize the advantage of some alternatives over the others when all compatible value functions acknowledge this advantage, or reduce the ambiguity in the advantage of some alternatives over the others when some compatible value functions acknowledge an advantage and other ones acknowledge a disadvantage. The basic procedure is refined by few extensions. They enable emphasizing the advantage of alternatives that could be considered as potential best options, accounting for intensities of preference, or obtaining a desired type of the marginal value functions.     

Profiling effects in industrial data mining by non-parametric DOE methods: An application on screening checkweighing systems in packaging operations

There is a growing interest in applying robust techniques for profiling complex processes in industry. In this work, we present an approach for analyzing fractional-factorial data by building distribution-free models suitable for dealing with replicated trials in search of non-linear effects. The technique outlined in this article is synthesized by implementing four key elements: (1) the data collection efficiency of non-linear fractional factorial designs, (2) the data compression capabilities of rank-sums for repetitive sampling schemes, (3) the rank-ordering as a means to transform data, and (4) the non-parametric screening for prominent effects where the normality and sparsity assumptions are waived. The technique is tested on four controlling factors for profiling the packaging weighing operations of a pharmaceutical enterprise. The robust data mining of repeated trials based on an L₉(3⁴) orthogonal array scheme with embedded uncontrolled noise is discussed extensively. The technique has been subjected to quality control as it is tested with well-defined artificial data. Concluding remarks involve contrasting this new technique with mainstream competing schemes.     

Portfolio value-at-risk optimization for asymmetrically distributed asset returns

We propose a new approach to portfolio optimization by separating asset return distributions into positive and negative half-spaces. The approach minimizes a newly-defined Partitioned Value-at-Risk (PVaR) risk measure by using half-space statistical information. Using simulated data, the PVaR approach always generates better risk-return tradeoffs in the optimal portfolios when compared to traditional Markowitz mean–variance approach. When using real financial data, our approach also outperforms the Markowitz approach in the risk-return tradeoff. Given that the PVaR measure is also a robust risk measure, our new approach can be very useful for optimal portfolio allocations when asset return distributions are asymmetrical.     

Perturbation and robust stability of autonomous singular linear matrix difference equations

In the perturbation theory of linear matrix difference equations, it is well known that the theory of finite and infinite elementary divisors of regular matrix pencils is complicated by the fact that arbitrarily small perturbations of the pencil can cause them to disappear. In this paper, the perturbation theory of complex Weierstrass canonical form for regular matrix pencils is investigated. By using matrix pencil theory and the Weierstrass canonical form of the pencil we obtain bounds for the finite elementary divisors of a perturbed pencil. Moreover we study robust stability of a class of linear matrix difference equations (of first and higher order) whose coefficients are square constant matrices.     

A robust robust optimization result

We study the loss in objective value when an inaccurate objective is optimized instead of the true one, and show that “on average” this loss is very small, for an arbitrary compact feasible region.     

Photonic analog-to-digital converter based on the robust symmetrical number system

A novel approach to realizing photonic analog-to-digital conversion with Gray-code-like property is proposed and demonstrated. Instead of using Mach-Zehnder modulators (MZMs) with different half-wave voltages, an array of MZMs with identical half-wave voltages are applied to realize quantization and encoding, which greatly simplifies the implementation. Multiple comparators with preset thresholds are applied at the output of each MZM to improve the number of bits. Through properly setting the bias voltages of the MZMs, a photonic analog to digital converter (ADC) based on the robust symmetrical number system (RSNS) coding method is realized. As an example, a 3-channel structure with maximum quantization level of 17 (corresponding to 4.09 bits) is investigated in detail. We show that the differential encoding technique can be applied in the proposed structure, which increases the equivalent number of bits of the ADC system.     

Reduced stability testing set for a diamond-type family of quasipolynomials

A diamond-type family of quasipolynomials, for which vertex stability results hold, is presented. Both delay independent and delay-dependent stability conditions are given. In the first case, it suffices to check for the stability of a finite testing set of multivariate polynomials. While in the other one, it is also needed to check the stability of several edge sub-families of quasipolynomials.

Global robust asymptotic stability analysis of BAM neural networks with time delay and impulse: An LMI approach

The global robust asymptotic stability of bi-directional associative memory (BAM) neural networks with constant or time-varying delays and impulse is studied. An approach combining the Lyapunov-Krasovskii functional with the linear matrix inequality (LMI) is taken to study the problem. Some a criteria for the global robust asymptotic stability, which gives information on the delay-dependent property, are derived. Some illustrative examples are given to demonstrate the effectiveness of the obtained results.     

Adaptive robust control of artificial swarm systems

We consider an artificial swarm system consisting of multi-agents. The agents may interact with each other based on their relative positions. Each agent exhibits a repulsion/attraction behavior toward another agent, which mimics some biological swarm systems. The performance of each individual agent is the accumulation of these respective considerations toward other agents. The overall performance of the artificial swarm system mimics the aggregation and formation in biological systems. We propose an adaptive robust control for each agent toward achieving the performance. The control can withstand uncertainty, which is time-varying, nonlinear, and without known bound. The controlled system converges to the desirable swarm system performance regardless of the uncertainty.