An explicit state-space solution to the one-block super-optimal distance problem

An explicit state-space approach is presented for solving the super-optimal Nehari-extension problem. The approach is based on the all-pass dilation technique developed in (Jaimoukha and Limebeer in SIAM J Control Optim 31(5):1115–1134, 1993) which offers considerable advantages compared to traditional methods relying on a diagonalisation procedure via a Schmidt pair of the Hankel operator associated with the problem. As a result, all derivations presented in this work rely only on simple linear-algebraic arguments. Further, when the simple structure of the one-block problem is taken into account, this approach leads to a detailed and complete state-space analysis which clearly illustrates the structure of the optimal solution and allows for the removal of all technical assumptions (minimality, multiplicity of largest Hankel singular value, positive-definiteness of the solutions of certain Riccati equations) made in previous work (Halikias et al. in SIAM J Control Optim 31(4):960–982, 1993; Limebeer et al. in Int J Control 50(6):2431–2466, 1989). The advantages of the approach are illustrated with a numerical example. Finally, the paper presents a short survey of super-optimization, the various techniques developed for its solution and some of its applications in the area of modern robust control.     

An efficient algorithm for the solution of the general mathematical programming problem

Conclusions As is obvious from these tables, the proposed algorithm allows problem (4), (6) to be solved fairly fast. At the same time, it depends weakly both on the dimensions of the problem and on the required precision of solution. This justifies reliance on the effectiveness of applying auxiliary problems, using nonlinear approximation of the constraints, in iterative methods of nonlinear programming.Translated from Kibernetika, No. 2, pp. 88–91, March–April, 1977.     

Dynamic programming solution to the batching problem in just-in-time flow-shops

Mixed-model manufacturing systems are widely used by companies, in order to meet the customers’ demand for a variety of products, in an efficient way. This paper is concerned with a special class of mixed-model manufacturing systems: flow-shops. In a flow-shop, each product has to be processed by a number of machines, following a common route. We study the production smoothing problem under presence of non-zero setup and processing times which also vary among the products. We split the master problem into two sub-problems which are concerned with determining the batch sizes and production sequences, respectively. We develop a dynamic programming procedure to solve the batching problem, and suggest using an existing method from the current literature to solve the sequencing problem. We conduct a computational study and show that our solution approach is effective in meeting the JIT goals and efficient in its computational requirements.     

A dynamic programming solution to a generalized LCS problem

In this paper, we consider a generalized longest common subsequence problem, the string-excluding constrained LCS problem. For the two input sequences X and Y of lengths n and m, and a constraint string P of length r, the problem is to find a common subsequence Z of X and Y excluding P as a substring and the length of Z is maximized. The problem and its solution were first proposed by Chen and Chao (2011) [1], but we found that their algorithm cannot solve the problem correctly. A new dynamic programming solution for the STR-EC-LCS problem is then presented in this paper. The correctness of the new algorithm is proved. The time complexity of the new algorithm is $O(nmr)$.     

A logarithmic-time solution to the point location problem for parametric linear programming

The optimiser of a (multi) parametric linear program (pLP) is a piecewise affine function defined over a polyhedral subdivision of the set of feasible states. Once this affine function has been pre-calculated, the optimal solution can be computed for a particular parameter by determining the region that contains it. This is the so-called point location problem. In this paper, we show that this problem can be written as an additively weighted nearest neighbour search that can be solved in time linear in the dimension of the state space and logarithmic in the number of regions.

It is well-known that linear model predictive control (MPC) problems based on linear control objectives (e.g., 1- or ∞-norm) can be posed as pLPs, and on-line calculation of the control law involves the solution to the point location problem. Several orders of magnitude sampling speed improvement are demonstrated over traditional MPC and closed-form MPC schemes using the proposed scheme.     

An options-based solution to the sequential auction problem

The sequential auction problem is commonplace in open, electronic marketplaces such as eBay. This is the problem where a buyer has no dominant strategy in bidding across multiple auctions when the buyer would have a simple, truth-revealing strategy if there was but a single auction event. Our model allows for multiple, distinct goods and market dynamics with buyers and sellers that arrive over time. Sellers each bring a single unit of a good to the market while buyers can have values on bundles of goods. We model each individual auction as a second-price (Vickrey) auction and propose an options-based, proxied solution to provide price and winner-determination coordination across auctions. While still allowing for temporally uncoordinated market participation, this options-based approach solves the sequential auction problem and provides truthful bidding as a weakly dominant strategy for buyers. An empirical study suggests that this coordination can enable a significant efficiency and revenue improvement over the current eBay market design, and highlights the effect on performance of complex buyer valuations (buyers with substitutes and complements valuations) and varying the market liquidity.     

An introduction to nonlinear programming—II: The linear programming problem

Linear programming problems represent the most thoroughly analyzed and widely solved class of parameter optimization problems. In Part II, we shall restrict our attention to this general class of problems. The characteristics of the admissible region are investigated and established. The Kuhn-Tucker conditions developed in Part I are applied to establish necessary and sufficient conditions that must be satisfied at a minimum. Included in the discussion is a consideration of dual linear programming problems. Then, we direct our attention to the question of determining the solution of specific problems. A general algorithm known as the Simplex Method is described and applied to several examples.     

An explicit solution to right factorization with application in eigenstructure assignment

1Proble mfor mulationRight factorization oflinear systems is a basic problemincontrol systems theory,and has important applications inmany analysis and design problems ,such as , robuststabilization [1] ,minimal state space realization [2,3] ,speed servo system design [4] ,synthesis of H_infinitycontrollers [5] ,etc .Furthermore ,it has been shown byDuan and his co_authors that a factorization can be usedtoparameterize all the solutions to a generalized_type ofSylvester matrix equations [6 ~8…     

An explicit scheme for the solution of the filtration problems

Different models of compressible fluid filtration are considered. Unlike the classical system of equations, the continuity equation is modified with allowance for the minimum scale of space averaging and for the internal relaxation time of the system. Three-level explicit finite difference schemes are proposed that are convenient for high-performance parallel implementation. The transition from the parabolic to the hyperbolic system of equations makes the stability requirements for them less stringent than for the two-level schemes.     

An algorithm for multi-parametric quadratic programming and explicit MPC solutions

Explicit solutions to constrained linear model predictive control problems can be obtained by solving multi-parametric quadratic programs (mp-QP) where the parameters are the components of the state vector. We study the properties of the polyhedral partition of the state space induced by the multi-parametric piecewise affine solution and propose a new mp-QP solver. Compared to existing algorithms, our approach adopts a different exploration strategy for subdividing the parameter space, avoiding unnecessary partitioning and QP problem solving, with a significant improvement of efficiency.     

An efficient solution to the five-point relative pose problem

An efficient algorithmic solution to the classical five-point relative pose problem is presented. The problem is to find the possible solutions for relative camera pose between two calibrated views given five corresponding points. The algorithm consists of computing the coefficients of a tenth degree polynomial in closed form and, subsequently, finding its roots. It is the first algorithm well-suited for numerical implementation that also corresponds to the inherent complexity of the problem. We investigate the numerical precision of the algorithm. We also study its performance under noise in minimal as well as overdetermined cases. The performance is compared to that of the well-known 8 and 7-point methods and a 6-point scheme. The algorithm is used in a robust hypothesize-and-test framework to estimate structure and motion in real-time with low delay. The real-time system uses solely visual input and has been demonstrated at major conferences.     

An alternative solution to the model structure selection problem

An alternative solution to the model structure selection problem is introduced by conducting a forward search through the many possible candidate model terms initially and then performing an exhaustive all subset model selection on the resulting model. An example is included to demonstrate that this approach leads to dynamically valid nonlinear models     

An optimal distributed solution to the dining philosophers problem

An optimal distributed solution to the dining philosophers problem is presented. The solution is optimal in the sense that it incurs the least communication and computational overhead, and allows the maximum achievable concurrency. The worst case upper bound for concurrency is shown to ben div 3,n being the number of philosophers. There is no previous algorithm known to achieve this bound.