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Finiteness of the Criss-Cross Algorithm for the Linear Programming Problem with S-Monotone Index Selection Rules

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Finiteness of the Criss-Cross Algorithm for the Linear Programming Problem with S-Monotone Index Selection Rules PU.M.A. Vol. 28 (2020), No. 1, pp. 1–13, DOI:10.1515/puma-2015-0041 Finiteness of the criss-cross algorithm for the linear programming problem with s-monotone index selection rules Adrienn Csizmadia∗† Marston Green, UK (Received: March 7, 2021) Abstract The traditional criss-cross algorithm for the linear programming prob- lem is shown to be finite when s-monotone index selection rules are used. The set of s-monotone index selection rules, among others, include the Last In First Out (LIFO) and the Most Often Selected Variable rule (MOSV). The advantage of applying the s-monotone index selection rule is the flexibility it provides in selecting the pivot element while still preserv- ing the guarantee for finiteness. Such flexibility may be used to improve the numerical stability of the algorithm. Mathematics Subject Classifications (2015). 90C49, 90C05 Keywords. Criss-cross method, index selection, finiteness 1 Introduction and problem definitions The linear programming problem (LP) is one of the most studied fields of oper- ations research. Most pivot algorithms (PA) for LPs minimize a merit function like the sum of infeasibilities or the objective function. An early exception, intro- duced by Terlaky [24, 25] and independently by Zionts [29] without a finiteness proof, has been the criss-cross algorithm (CCA) in which the proof of finiteness depends on purely combinatorial considerations as opposed to simplex type methods that are always finite if the problem is non-degenerate. This reliance on index selection rules of the criss-cross algorithm makes studying the class of index selection rules for which it is finite and allows flexible pivot selection of particular interest. The scope of the method has been extensively broadened in [1, 2, 13, 18, 21, 26]. In linear programming, an often cited way for selecting the pivot position is using the minimal index rule. Though such a pivot position is good in theory it may not necessarily be suitable in practice. It would be interesting to use a pivot rule which provides finiteness but also offers more flexibility. To allow ∗Maiden name Adrienn Nagy †e-mail: [email protected] 1 2 CSIZMADIA that, the idea of Zhang was to use the Most-Often-Selected-Variable and Last- In-First-Out rules to prove the finiteness of the criss-cross algorithm. The same is proven using the Orthogonality theorem in [20]. This paper shows that the recently introduced concept of s-monotone index selection rules [9, 10, 15, 16] can also be applied to guarantee the finiteness of the criss-cross algorithm. This proof first appeared as part of the PhD dissertation [23] of the author. Throughout the paper, matrices are denoted by italic capital letters, vectors by bold, scalars by normal letters and index sets by capital calligraphic letters. Columns of a matrix are indexed as a subscript while rows are indexed by superscripts. A 2 Rm×n and M 2 Rn×n denote the original problem matrices for various problem types, while b and c denote the right hand side and objective vectors, respectively. Let A 2 IRm×n; c 2 IRn; b 2 IRm be a matrix and vectors of appropriate dimensions, then min cT x Ax = b (1) x ≥ 0 is a primal linear programming (P-LP) problem, while the dual linear program- ming (D-LP) problem can be defined as follows max bT y (2) AT y ≤ c where x 2 IRn and y 2 IRm are primal and dual decision vectors, respectively. Without loss of generality we assume that rank(A) = m. A regular m × m rectangular submatrix of the constraint matrix is called a basis [15]. For a given basis B, we denote the nonbasic part of A by N; the corre- sponding set of indices for the basic and nonbasic parts are denoted by IB and IN respectively. The corresponding short pivot tableau for B is denoted by T := B−1N, while the transformed right hand side and objective vectors are denoted by b := B−1b and c := cT B−1. We refer to the rows (index i) and columns (index j) of the short pivot tableau corresponding to a given basis −1 (i) −1 (i) B as tj := B aj and t := (B N) [20]. We will denote the individual coefficients of the pivot tableau as tij . The variables corresponding to the column vectors of the basis B are called basic variables. (i) Define (column) vectors t and tj with dimension (n+2) [20], corresponding to the (primal) basic tableau of the (P-LP) problem, where i 2 IB and j 2 IN , respectively, in the following way: 8 < tik if k 2 IB [IN (i) ¯ (t )k = tik = : bi if k = b (3) 0 if k = c FINITENESS OF THE CRISS-CROSS ALGORITHM. 3 and 8 > 2 I <> tkj if k B −1 if k = j (tj)k = tkj = > 2 I n f g [ f g (4) :> 0 if k ( N j ) b c¯j if k = c where b and c denotes indices associated with vectors b and c, respectively. (c) Furthermore, define t and tb vectors in the following way 8 < c¯k if k 2 IB [IN (c) (t )k = tck = : 1 if k = c (5) − T −1 cBB b if k = b and 8 > ¯ 2 I <> bk if k B −1 if k = b (tb)k = tkb = > 2 I (6) :> 0 if k N − T −1 cBB b if k = c and from now on we assume that c is always a basic index, while b is always a nonbasic index of the (P-LP) problem. The following result is widely used in the proofs of finiteness of pivot algorithms for LP. Theorem 1 [20] Let a (P-LP) problem be given, with rank(A) = m and assume that IB0 and IB00 are two arbitrary bases of the problem. Then 00(i) T 0 (t ) tj = 0 (7) for all i 2 IB00 and for all j 62 IB0 . 2 The s-monotone index selection rules The s-monotone index selection rules were introduced in [9, 10] for LP and LCP problems, and later directly applied to QPs in [8]. Pivot based algorithms (like the simplex algorithm [11], MBU-simplex algo- rithm [3] or the criss–cross algorithm [24, 25, 29]) often feature the following similar principles: 1. The main flow of the algorithm is defined by a pivot selection rule which defines the basic characteristics of the algorithm, though the pivot position defined by it is not necessarily unique (see for instance [7, 12, 22]), a series of "wrong" choices may even lead to cycling [7, 22]. 2. To avoid the possibility of cycling, an index selection rule is used as an anti-cycling strategy (see for instance [6, 7, 27]), which may be flexible [9, 16] but usually at several bases during the algorithm, it defines the pivot position uniquely. 4 CSIZMADIA For several pivot algorithms, – like simplex, MBU-simplex or criss–cross algorithms – proofs of finiteness are often based on the orthogonality theorem [5, 15, 16, 20], considering a minimal cycling example [5, 15, 16, 10]. In minimal cycling examples, all variables should move during a cycle – if such exists – and following the movements of the least preferred variable according to the index selection rule [6, 9, 10, 15, 16, 19, 28], using orthogonality theorem we obtain contradiction. Examples of such pivot and index selection rules include 1. Pivot selection rules for (P-LP): (a) simplex [11] (Pivot column selection: negative reduced cost. Pivot element selection: using ratio test. Preserving non negativity of the right hand side.) (b) MBU simplex [3] (Pivot column selection: negative reduced cost, choosing driving variable. Pivot element selection: defining driving and auxiliary pivots using primal and after that dual ratio tests. Monotone in the reduced cost of the driving variable.) (c) criss–cross [25] (Pivot column/row selection is based on infeasibility – negative right hand side or negative reduced cost. Pivot element selection: admissible pivot positions.) 2. Index selection rules: (a) Bland’s or the minimal index rule [6] (b) Last-In-First-Out (LIFO) (c) Most-Often-Selected-Variable (MOSV) LIFO and MOSV index selection rules for linear programming problems were first used by S. Zhang [28] to prove the finiteness of the criss–cross algorithm with these anti-cycling index selection rules. Bilen, Csizmadia and Illés [5] proved that variants of MBU simplex algorithm are finite with both LIFO and MOSV index selection rules, while Csizmadia in his PhD Thesis [9] and Csiz- madia et al [10] showed that the simplex algorithm is finite when the LIFO and MOSV are applied. These results led to the joint generalization of the above mentioned anti-cycling index selection rules. The following general framework for proving the finiteness of several pivot algorithms and index selection rule combinations is introduced, as in [10]. Definition 1 (Possible pivot sequence [10]) A sequence of index pairs S = fSk = (ik; ok): ik; ok 2 IN for some consecutive k 2 INg; is called a possible pivot sequence, if (i) n = maxfmax ik; max okg is finite, k2IN k2IN (ii) there exists a (P-LP) with n variables and rank(A) = m, and FINITENESS OF THE CRISS-CROSS ALGORITHM. 5 (iii) the (possibly infinite) pivot sequence is such that the moving variable pairs of (P-LP) correspond to the index pairs of S. The index pairs of a possible pivot sequence are thus only required to comply with the basic and nonbasic status. It is now easy to show that Proposition 1 If a possible pivot sequence is not finite then there exists a (sub)set of indices, I∗, that occur infinitely many times in S: Definition 2 (Pivot index preference [9, 10]) A sequence of vectors sk 2 Nn is called a pivot index preference of an index selection rule, if in iteration j, in the case of ambiguity according to a pivot selection rule, the index selection rule selects an index with highest value in sj among the candidates.
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