Cylindrical Algebraic Decomposition Method Research Articles

Projection-based robust optimization with symbolic computation

Robust counterpart reformulation is a common technique used to deal with data uncertainty in robust optimization (RO) problems. The derivation of the robust counterpart formulation using the duality theory is nontrivial, especially for complex uncertainty sets. To reduce the dependence on robust counterparts, a novel method is proposed in this article for RO problems. Based on the feasible space projection, the proposed method can locate robust solutions without formulating the robust counterparts. RO model can be reformulated as a semi-algebraic system and a modified cylindrical algebraic decomposition method is applied to project the high-dimensional feasible space on the low-dimensional space of the objective function and uncertainty parameters. By solving the maximization problem and using the max-max decision criterion, the final robust solution can be selected. The case studies, involving problems of nonlinear programming (NLP) and robust design optimization problems, show that the proposed method can obtain the robust solution effectively.

Analytical solution of volumetric flexibility through symbolic computation

Volumetric flexibility index is an indicator for describing the flexibility of a process design. However, there is no prior literature on how the theoretical value of the volumetric flexibility index can be obtained. In this work, we propose the first systematic method to accurately derive the analytical solution of the volumetric flexibility index through symbolic computation. First, based on the cylindrical algebraic decomposition method, the feasible space can be decomposed into finite valid cells with analytical expressions. Then, the integral operation is implemented to calculate the hypervolume for each cell. Finally, hypervolumes of all feasible cells are summed to determine the volumetric flexibility index. Case studies show that the proposed method can obtain the theoretical value of the volumetric flexibility index for nonconvex systems. In addition, the explicit information between the hypervolume and design variables can be derived, which can be further applied to hypervolume optimization problems under different designs.

Fully incremental cylindrical algebraic decomposition

Collins introduced the cylindrical algebraic decomposition method for eliminating quantifiers in real arithmetic formulas. In our work we use this method for satisfiability checking in satisfiability modulo theories solver technologies, and tune it by trying to avoid some computation steps that are needed for quantifier elimination but not for satisfiability checking. We further propose novel data structures and adapt the method to work incrementally and to support backtracking. We verify the effectiveness experimentally by comparing different versions of the cylindrical algebraic decomposition used within a state-of-the-art satisfiability modulo theories solver.

Robust bounds for fractional-order systems with uncertain order and structured perturbations via Cylindrical Algebraic Decomposition method

This paper considers the robust stability problem of fractional-order systems with uncertain order and structured perturbations. A stability check procedure is proposed for determining the robust bounds of uncertain order and other uncertain parameters for fractional-order systems.The results are obtained in terms of Cylindrical Algebraic Decomposition which is first used for analyzing the robust stability problem of fractional-order systems with uncertain order. The method is non-conservative for fractional-order systems with the uncertain order α satisfying 0 < α < 2. Examples are given to demonstrate the effectiveness of proposed approach.

Influence of design parameters on the singularities and workspace of a 3-RPS parallel robot

This paper presents variations in the workspace, singularities, and joint space with respect to design parameter k, which is the ratio of the dimensions of the mobile platform to the dimensions of the base of a 3-RPS parallel manipulator. The influence of the design parameters on parasitic motion, which is important when selecting a manipulator for a desired task, is also studied. The cylindrical algebraic decomposition method and Gröbner-based computations are used to model the workspace and joint space with parallel singularities in 2R1T (two rotational and one translational) and 3T (three translational) projection spaces, where the orientation of the mobile platform is represented using quaternions. These computations are useful in selecting the optimum value for the design parameter k such that the parasitic motions can be limited to specific values. Three designs of the 3-RPS parallel robot, based on different values of k, are analyzed.

Real zeros of the zero-dimensional parametric piecewise algebraic variety

The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex.

BEAMS ON TENSIONLESS ELASTIC FOUNDATION: APPROXIMATE QUANTIFIER ELIMINATION WITH CHEBYSHEV SERIES

The well-known Sturm's theorem (based on Sturm's sequences) for the determination of the number of distinct real zeros of polynomials in a finite or infinite real interval has been already used in elementary quantifier elimination problems including applied mechanics and elasticity problems. Here it is further suggested that this theorem can also be used for quantifier elimination, but in more complicated problems where the functions involved are not simply polynomials, but they may contain arbitrary transcendental functions. In this case, it is suggested that the related transcendental equations/inequalities can be numerically approximated by polynomial equations/inequalities with the help of Chebyshev series expansions in numerical analysis. The classical problem of a straight isotropic elastic beam on a tensionless elastic foundation, where the deflection function (incorporating both the exponential function and trigonometric functions) should be continuously positive (this giving rise to a quantifier elimination problem along the length of the beam) is used as an appropriate vehicle for the illustration of the present mixed (symbolic–numerical) approach. Two such elementary beam problems are considered in some detail (with the help of the Maple V computer algebra system) and the related simple quantifier-free formulae are established and seen to coincide with those already available in the literature for the same beam problems. More complicated problems, probably necessitating the use of more advanced computer algebra techniques (together with Sturm's theorem), such as the Collins well-known and powerful cylindrical algebraic decomposition method for quantifier elimination, can also easily be employed in the present approximate (because of the use of Chebyshev series expansions) symbolic–numerical computational environment.

Computer-aided quantifier elimination in crack problems under constraints for the stress intensity factors

It is suggested that crack problems under constraints, concerning the values of the parameters involved, be studied by the method of quantifier elimination in such a way that quantifier-free formulae can be derived assuring the validity of a restriction about the stress intensity factors (usually that such a factor does not exceed a critical value) for all possible values of the parameters under the applicable constraints. This permits safe conclusions about the possible fracture of a cracked specimen (under constraints on the parameters involved) for every set of values of the geometry/loading/fracture parameters. The elementary crack problem of a single straight crack in an infinite plane isotropic elastic medium is used as the vehicle for the illustration of the present approach and the related quantifier-free formula is easily derived. A simple case of a periodic array of cracks is also considered in brief. The classical algebraic tools of resultants and Sturm's sequences (together with Sturm's theorem) as well as the computer algebra system Maple V have been used in the present computations. More general results, with the help of Collins' famous cylindrical algebraic decomposition method, in more complicated crack problems, can also be obtained.

Partial Cylindrical Algebraic Decomposition for quantifier elimination

The Cylindrical Algebraic Decomposition method (CAD) decomposes Rr into regions over which given polynomials have constant signs. An important application of CAD is quantifier elimination in elementary algebra and geometry. In this paper we present a method which intermingles CAD construction with truth evaluation so that parts of the CAD are constructed only as needed to further truth evaluation and aborts CAD construction as soon as no more truth evaluation is needed. The truth evaluation utilizes in an essential way any quantifiers which are present and additionally takes account of atomic formulas from which some variables are absent. Preliminary observations show that the new method is always more efficient than the original, and often significantly more efficient.

Algebraic decomposition of regular curves

The cylindrical algebraic decomposition method decomposes E r into regions over which a given polynomial has constant sign by extension of one complicated decomposition of E r-1 . We investigate a method which decomposes E r into sign-invariant region by combining several but simpler decompositions of E r-1 . We can obtain a sign-invariaat decomposition of E 2 defined by a bivariate polynomial of total degree n and coefficient size d in time O(n 12 (d + log n) 2 log n) . Preliminary experiments suggest that the method is useful in practice.