Optimization of quadratic forms and t -norm forms on interval - PowerPoint PPT Presentation

Optimization of quadratic forms and t -norm forms on interval domains and computational complexity ık 2 and Vladik Kreinovich 3 Michal ˇ y 1 , Milan Hlad´ Cern´ 1 University of Economics, Prague, Czech Republic 2 Charles University, Prague, Czech Republic 3 University of Texas at El Paso, Texas, USA 7th World Conference on Soft Computing, Baku, 2018 M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 1 / 16

Problem formulation � n � n i =1 f i ( x i )+ � n i =1 f i ( x i ) i � = j g ij ( x i , x j ) F F · · · · · · x 1 x 2 x 3 x n x 1 x 2 x 3 x n separable case quadratic interactions M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 2 / 16

Problem formulation (contd.) Model for imprecision of inputs. Instead of x i we can observe only bounds x i , x i s.t. x i � x i � x i , i = 1 , . . . , n . M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 3 / 16

Problem formulation (contd.) Model for imprecision of inputs. Instead of x i we can observe only bounds x i , x i s.t. x i � x i � x i , i = 1 , . . . , n . The main question. Given F : R n → R , can we find bounds on F ( x 1 , . . . , x n ) given the observable bounds [ x 1 , x 1 ] , . . . , [ x n , x n ]? More formally. The task is to compute F = max { F ( x 1 , . . . , x n ) | x i ∈ [ x i , x i ] , i = 1 , . . . , n } , F = min { F ( x 1 , . . . , x n ) | x i ∈ [ x i , x i ] , i = 1 , . . . , n } . M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 3 / 16

Some well-known results Theorem (the general case). For a general function F , the bounds F , F are nonrecursive. M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 4 / 16

Some well-known results Theorem (the general case). For a general function F , the bounds F , F are nonrecursive. Observation (the separable case). For the separable case n � F ( x 1 , . . . , x n ) = f i ( x i ) , i =1 the bounds reduce to n n � � F = f i ( x i ) , F = f i ( x i ) , i =1 i =1 where f i = min x i � ξ � x i f i ( ξ ) and f i = max x i � ξ � x i f i ( ξ ). M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 4 / 16

The case with quadratic interactions M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 5 / 16

The case with quadratic interactions The general form: F = � i f i ( x i )+ � j � = i g ij ( x i , x j ) M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 5 / 16

The case with quadratic interactions The general form: F = � i f i ( x i )+ � j � = i g ij ( x i , x j ) A natural example — a quadratic form: � � q ii x 2 q ij x i x j = x T Qx , F = i + i j � = i where Q = ( q ij ) i , j =1 ,..., n and x = ( x 1 , . . . , x n ) T . M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 5 / 16

The case with quadratic interactions The general form: F = � i f i ( x i )+ � j � = i g ij ( x i , x j ) A natural example — a quadratic form: � � q ii x 2 q ij x i x j = x T Qx , F = i + i j � = i where Q = ( q ij ) i , j =1 ,..., n and x = ( x 1 , . . . , x n ) T . Classical result: If Q is psd, then F is computable in polynomial time (via IPMs for convex quadratic programming). M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 5 / 16

The case with quadratic interactions The general form: F = � i f i ( x i )+ � j � = i g ij ( x i , x j ) A natural example — a quadratic form: � � q ii x 2 q ij x i x j = x T Qx , F = i + i j � = i where Q = ( q ij ) i , j =1 ,..., n and x = ( x 1 , . . . , x n ) T . Classical result: If Q is psd, then F is computable in polynomial time (via IPMs for convex quadratic programming). On the contrary: For Q psd, computation of F is an NP-hard problem . M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 5 / 16

The case with quadratic interactions The general form: F = � i f i ( x i )+ � j � = i g ij ( x i , x j ) A natural example — a quadratic form: � � q ii x 2 q ij x i x j = x T Qx , F = i + i j � = i where Q = ( q ij ) i , j =1 ,..., n and x = ( x 1 , . . . , x n ) T . Classical result: If Q is psd, then F is computable in polynomial time (via IPMs for convex quadratic programming). On the contrary: For Q psd, computation of F is an NP-hard problem . Our general goal: Inspect further complexity-theoretic results. M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 5 / 16

Sparse quadratic forms Definition. The quadratic form F ( x ) = x T Qx is sparse is there are “many” zeros in the matrix Q . M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 6 / 16

Sparse quadratic forms Definition. The quadratic form F ( x ) = x T Qx is sparse is there are “many” zeros in the matrix Q . Problem. How “sparse” must the matrix Q be to make the values F , F polynomially computable? M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 6 / 16

Sparse quadratic forms Definition. The quadratic form F ( x ) = x T Qx is sparse is there are “many” zeros in the matrix Q . Problem. How “sparse” must the matrix Q be to make the values F , F polynomially computable? Solution. Theorem 1. If there are at most O (log n ) nonzero off-diagonal entries in Q , then both bounds F , F are computable in polynomial time. M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 6 / 16

Sparse quadratic forms Definition. The quadratic form F ( x ) = x T Qx is sparse is there are “many” zeros in the matrix Q . Problem. How “sparse” must the matrix Q be to make the values F , F polynomially computable? Solution. Theorem 1. If there are at most O (log n ) nonzero off-diagonal entries in Q , then both bounds F , F are computable in polynomial time. Theorem 2. But: Even if the matrix Q is psd and there are Ω( n ε ) non-zero off-diagonal entries in Q , for an arbitrarily small ε > 0, then computation of F is NP-hard. M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 6 / 16

(In)approximability Let us refine the results: M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 7 / 16

(In)approximability Let us refine the results: Corollary (absolute approximation). When we have Ω( n ε ) non-zero off-diagonal entries, then F is inapproximable with an arbitrarily large absolute error. Relative approximation. A result by Nesterov (1998) implies that the problem can be approximated with some “reasonable” relative error by semidefinite relaxation. M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 7 / 16

The quadratic form as a graph The Q -graph of a quadratic form x T Qx : Assume that Q is triangular (without loss of generality). Define the Q -graph as follows: vertices: variables x 1 , . . . , x n , edges: { x i , x j } is an edge if i � = j and q ij � = 0, weights of edges: the weight of the edge is q ij . x 1 x 2 2 0 1 1 q 11 1 0 3 q 22 x 5 1 1 Q = q 33 x 3 2 q 44 x 4 q 55 M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 8 / 16

Sunflower graph Special shapes of the Q -graph. . . Sunflower graph G : There exists a cut C of vertex size O (log n ) such that G \ C has components of vertex size O (log n ). Theorem. If the Q -graph is a sunflower graph, then F is computable in polynomial time. Component 1 Component 2 · · · x 1 x 8 cut x 3 · · · Component 4 Component 3 M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 9 / 16

Further special forms of Q -graphs Further polynomially solvable cases: Q -graph is a tree M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 10 / 16

Further special forms of Q -graphs Further polynomially solvable cases: Q -graph is a tree Q -graph is planar, with O (log n ) faces M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 10 / 16

Further special forms of Q -graphs Further polynomially solvable cases: Q -graph is a tree Q -graph is planar, with O (log n ) faces Q -graph is a bipartite graph, where one of the partites has size O (log n ) M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 10 / 16

Further special forms of Q -graphs Further polynomially solvable cases: Q -graph is a tree Q -graph is planar, with O (log n ) faces Q -graph is a bipartite graph, where one of the partites has size O (log n ) (few negative coefficients) there exists a cut C of size O (log n ), such that all variables incident with negative coefficients are in C M.ˇ C., M.H., V.K. (Prague & El Paso) Optimization of quadratic forms. . . WCONSC 2018, Baku 10 / 16