An Algorithm for Solving the Minimum-Norm Point Problem over the Intersection of a Polytope and an Affine Set

This chapter investigates the objects and mappings of algebraic geometry from a geometric point of view, making use especially of the algebraic tools of Chapter VII and of Sections 7–10 of Chapter VIII. In Sections 1–12, k denotes a fixed algebraically closed field. Sections 1–6 establish the definitions and elementary properties of varieties, maps between varieties, and dimension, all over k. Sections 1–3 concern varieties and dimension. Affine algebraic sets, affine varieties, and the Zariski topology on affine space are introduced in Section 1, and projective algebraic sets and projective varieties are introduced in Section 3. Section 2 defines the geometric dimension of an affine algebraic set, relating the notion to Krull dimension and transcendence degree. The actual context of Section 2 is a Noetherian topological space, the Zariski topology on affine space being an example. In such a space every closed subset is the finite union of irreducible closed subsets, and the union can be written in a certainway thatmakes the decomposition unique. Every nonempty closed set has a meaningful geometric dimension. In affine space the irreducible closed sets are the varieties, and each variety acquires a geometric dimension. The discussion in Section 2 applies in the context of projective space as well, and thus each projective variety acquires a geometric dimension. Moreover, any nonempty open subset of a Noetherian space is Noetherian. A nonempty open subset of an affine variety is called quasi-affine, and a nonempty open subset of a projective variety is called quasiprojective. Each quasi-affine variety or quasiprojective variety has a dimension equal to that of its closure, which is a variety. Sections 4–6 take up maps between varieties. Section 4 introduces spaces of scalar-valued functions on quasiprojective varieties–rational functions, functions regular at a point, and functions regular on an open set. The section goes on to relate these notions for the different kinds of varieties. Section 5 introduces morphisms, which are a restricted kind of function between varieties. The tools of Sections 4–5 together show that for many purposes all the different kinds of varieties can be treated as quasiprojective varieties. Section 6 introduces rational maps between varieties; these are not everywhere-defined functions, but each can be restricted to an open dense subset on which it is a morphism. Rational maps with dense image correspond to field mappings of the fields of rational functions, with the order of the mappings reversed. Section 7 concerns singularities at points of varieties, still over the field k. Zariski's Theorem was stated in Chapter VII for affine varieties and partly proved at that time. In the current context it has a meaning for any point of any quasiprojective variety. The section proves the full theorem, which characterizes singular points in a way that shows they remain singular under isomorphisms of varieties. Section 8 concerns classification questions over k for irreducible curves, i.e., quasiprojective varieties of dimension 1. From Section 6 it is known that two irreducible curves are equivalent under rational maps if and only if their fields of rational functions are isomorphic. The main theorem of Section 8 is that each such equivalence class of irreducible curves contains an everywhere nonsingular projective curve, and this curve is unique up to isomorphism of varieties. The points of this curve are parametrized by those discrete valuations of the underlying function field that are defined over k. Sections 9–12 relate the general theory of Sections 1–6 to the topic of solutions of simultaneous solutions of polynomial equations, as treated at length in Chapter VIII. Section 9 treats monomial ideals in k[X1, . . . , Xn], identifying their zero loci concretely and computing their dimension. The section goes on to introduce the affineHilbert function of this ideal, which measures the proportion of polynomials of degree≤ s not in the ideal. In the way that this function is defined, it is a polynomial for large s called the affine Hilbert polynomial of the ideal. Its degree equals the dimension of the zero locus of the ideal. Section 10 extends this theory from monomial ideals to all ideals, again concretely computing the dimension of the zero loci, obtaining an affine Hilbert polynomial, and showing that its degree equals the dimension of the zero locus of the ideal. Section 11 adapts the theory to homogeneous ideals and projective algebraic sets by making use of the cone in affine space over the set in projective space. Section 12 applies the theory of Section 11 to address the question how the dimension of a projective algebraic set is cut down when the set is intersected with a projective hypersurface. A consequence of the theory is the result that a homogeneous system of polynomial equations over an algebraically closed field with more unknowns than equations has a nonzero solution. Section 13 is a brief introduction to the theory of schemes, which extends the theory of varieties by replacing the underlying algebraically closed field by an arbitrary commutative ring with identity.

This paper presents a simple and efficient method based on artificial neural network to solve distributed optimal control of Poisson’s equation with Dirichlet boundary condition. The trial solutions are used to approximate the state and control variables. These trial solutions are considered by using a single layer neural network. By replacing the trial solutions in objective function and Poisson’s equation, then using the weighted residual method, distributed optimal control of Poisson’s equation is converted to a linear quadratic optimal control problem. The weights of the trial solutions are computed by solving the new problem. In order to solve the linear quadratic optimal control problem, the Pontryagin maximum principle is used. Finally we apply the proposed method on several examples that in computational experiments, the high efficiency of the presented method is illustrated.

The modern theory of condition measures for convex optimization problems was initially developed for convex problems in the conic format \[ (\mathrm{CP}_d)\qquad \ z_*:= \min_x \{c^t x | Ax-b \in C_Y, x \in C_X \}, \] and severalaspects of the theory have now been extended to handle nonconic formats as well. In this theory, the (Renegar) condition measure $C(d)$ for $({\rm CP}_d)$ has been shown to be connected to bounds on a wide variety of behavioral and computational characteristics of $({\rm CP}_d)$, from sizes of optimal solutions to the complexity of algorithms for solving $({\rm CP}_d)$. Herein we test the practical relevance of the condition measure theory, as applied to linear optimization problems that one might typically encounter in practice. Using the NETLIB suite of linear optimization problems as a test bed, we found that 71% of the NETLIB suite problem instances have infinite condition measure. In order to examine condition measures of the problems that are the actual input to a modern interior-point-method (IPM) solver, we also computed condition measures for the NETLIB suite problems after preprocessing by CPLEX 7.1. Here we found that 19% of the postprocessed problem instances in the NETLIB suite have infinite condition measure, and that log C(d) of the postprocessed problems is fairly nicely distributed. Furthermore, among those problem instances with finite condition measure after preprocessing, there is a positive linear relationship between IPM iterations and log C(d) of the postprocessed problem instances (significant at the 95% confidence level), and 42% of the variation in IPM iterations among these NETLIB suite problem instances is accounted for by log C(d) of the postprocessed problem instances.

We consider a real affine space X of finite dimension (which is always denoted by n), and whose underlying vector subspace \(\vec X\) (see 2.A) is endowed with a Euclidean structure; we say that X is a Euclidean affine space. The standard example is R n , considered as an affine space.

In this paper, we introduce the concept of trade-off ratio function, which is closely related to the well-known Geoffrion’s proper efficiency for multi-objective optimization problems, and investigate its boundedness property. For linear multi-objective optimization problems, we show that the trade-off ratio function is bounded on the efficient solution set. For piecewise linear multi-objective optimization problems, we show that all efficient solutions are always properly efficient in Borwein’s sense, and moreover, all efficient solutions are properly efficient in Geoffrion’s sense if and only if a recession condition holds. Finally, we provide an example to illustrate that the trade-off ratio function may be unbounded on the efficient solution set to piecewise linear multi-objective optimization problems, even if the recession condition holds, while it is bounded on the supported efficient solution set.

In this paper, the matrix game based on triangular intuitionistic fuzzy payoff is put forward. Then, we get a conclusion that the equilibrium solution of this game model is equivalent to the solution of a pair of the primal–dual single objective intuitionistic fuzzy linear optimization problems ( I F L O P 1 ) and ( I F L O D 1 ) . Furthermore, by applying the accuracy function, which is linear, we transform the primal–dual single objective intuitionistic fuzzy linear optimization problems ( I F L O P 1 ) and ( I F L O D 1 ) into the primal–dual discrete linear optimization problems ( G L O P 1 ) and ( G L O D 1 ) . The above primal–dual pair ( G L O P 1 ) – ( G L O D 1 ) is symmetric in the sense the dual of ( G L O D 1 ) is ( G L O P 1 ) . Thus the primal–dual discrete linear optimization problems ( G L O P 1 ) and ( G L O D 1 ) are called the symmetric primal–dual discrete linear optimization problems. Finally, the technique is illustrated by an example.

This paper proposes a novel and computationally efficient global optimization method based on swarm intelligence for locating nodes in a WSN environment. The mean squared range error of all neighbouring anchor nodes is taken as the objective function for this non linear optimization problem. The Particle Swarm Optimization (PSO) is a high performance stochastic global optimization tool that ensures the minimization of the objective function, without being trapped into local optima. The easy implementation and low memory requirement features of PSO make it suitable for highly resource constrained WSN environments. Computational experiments on data drawn from simulated WSNs show better convergence characteristics than the existing Simulated Annealing based WSN localization.

In several previous works the construction of a computationally hard function with respect to a certain class of algorithms or Boolean circuits has been used to derive small pseudo-random spaces. In this paper, we revert this connection by presenting two new direct relations between the efficient construction of pseudo-random (both two-sided and one-sided) sets for Boolean affine spaces and the explicit construction of Boolean functions having hard branching program complexityIn the case of 1-read branching programs (1-Br.Pr.), we show that the construction of non trivial (i.e. of cardinality 2o(n)) discrepancy sets (i.e. two-sided pseudo-random sets) for Boolean affine spaces of dimension greater than n/2 yield a set of explicit Boolean functions having very hard 1-Br.Pr. size. By combining the best known construction of ∈-biased sample spaces for linear tests and a simple “Reduction” Lemma, we derive the required discrepancy set and obtain a Boolean function in P having 1-Br.Pr. size not smaller than \(2^{n - O\left( {log^2 n} \right)} \) and a Boolean function in DTIME\(\left( {2^{O\left( {log^2 n} \right)} } \right)\) having 1-Br.Pr. size not smaller than 2n-O(log n). The latter bound is optimal and both of them are exponential improvements over the best previously known lower bound that was \(2^{n - 3n^{1/2} } \)[21].As for non deterministic syntactic k-read branching programs (k-Br.Pr.), we introduce a new method to derive explicit, exponential lower bounds that involves the construction of hitting sets (one-sided pseudo-random sets) for affine spaces of dimension o(n=2). Using an appropriate “orthogonal” representation of small Boolean affine spaces, we efficiently construct these hitting sets thus obtaining an explicit Boolean function in P that has k-Br.Pr. size not smaller than \(2^{n^{1 - o\left( 1 \right)} } \) for any \(k = o\left( {\frac{{\log n}}{{\log \log n}}} \right)\). This improves over the previous best known lower bounds given in [8,11,17] for some range of k

The object of the study is the technique for the formation of characteristic functions and Balyuba-Naidysh coordinates (BN-coordinates) in the composite method of geometric modeling. The existing methods of modeling economic, technological and any other processes on real objects are rather complicated, with significant limitations on the number of incoming factors. One of the most problematic places is the complexity and narrow scope of each of the existing modeling methods, restrains their distribution and practical implementation on real business entities. Hence the need to develop a universal method for modeling multifactor systems. The closest to this is the composite method of geometric modeling (CMGM), the universality of which is ensured, first of all, thanks to the use of the intrinsic technique of generating characteristic functions and BN-coordinates. The use of the Balyuba curves (B-curves) constructed in BN-coordinates in KMGM gives significant advantages to CMGM. One-, two-, three-parameter B-curve can be considered in the n-dimensional Euclidean space E n . As a result, CMGM can be used to solve problems in n-dimensional space, and the result of the solution can be decomposed into n one-dimensional projections, on which it is easy to analyze the solution. This can be used, in particular, in information systems to support management decisions. A feature of the B-curves is that the BN-coordinates p(t); q(t); r(t) are its parametric model, which is a constant. Applying a variety of variants of changing points, it is possible to obtain a lot of variants of B-curves, which is important for carrying out computer experiments in order to increase the adequacy of the constructed geometric model. The principle of the formation of characteristic functions is the operation of multiplying parameters and artificially assigned coefficients. As a result of the definition the product at the node points becomes zero or one, and in the intervals between nodal points it changes from zero to one. The number of factors of the characteristic function is equal to the number of nodal points that the characteristic function interpolates. The BN-coordinates of one B-curve form a system of interrelated fractional-rational functions. Thus, a generalizing technique for the algebraic formation of characteristic functions has been developed, and a transition from characteristic functions to BN-coordinates for the interpolation of three points has been determined. The technique used here can also be used for geometric interpolation of four or more points. The possibility of increasing the number of initial points of a geometric figure for BN-interpolation, extends the capabilities of models of multifactor processes and systems.

In 0.5.3 we have defined the real Euclidean space R n as the affine space E n whose vectors satisfy axioms E.1° – 5°. Since the inner product x 2 of a vector x = x 2 e i of this space by itself is a positive definite quadratic form x 2 = e ij x i x j , this space is also called a quadratic Euclidean space. In 0.5.3 we have seen that the space R n also satisfies the axioms M.1° – 3° of a metric space.

Let X be a subset of n points of the Euclidean space, and let 0 < e < 1. A classical result of Johnson and Lindenstrauss [JL] states that there is a projection of X onto a subspace of dimension O(e-2 log n) with distortion ≤ 1+ e. We show a natural extension of the above result to a stronger preservation of the geometry of finite spaces. By a k-fold increase of the number of dimensions used compared with [JL], a good preservation of volumes and of distances between points and affine spaces is achieved. Specifically, we show how to embed a subset of size n of the Euclidean space into a O(e-2 log n)-dimensional Euclidean space, so that no set of size s ≤ k changes its volume by more than (1 + es-1. Moreover, distances of points from affine hulls of sets of at most k - 1 points in the space do not change by more than a factor of 1 + e. A consequence of the above with k = 3 is that angles can be preserved using asymptotically the same number of dimensions as the one used in [JL]. Our method can be applied to many problems with high-dimensional nature such as Projective Clustering and Approximated Nearest Affine Neighbour Search. In particular, it shows a first polylogarithmic query time approximation algorithm to the latter. We also show a structural application that for volume respecting embedding in the sense introduced by Feige [F], the host space need not generally be of dimensionality greater than polylogarithmic in the size of the graph.

Compressive sensing relies on the sparse prior imposed on the signal of interest to solve the ill-posed recovery problem in an under-determined linear system. The objective function used to enforce the sparse prior information should be both effective and easily optimizable. Motivated by the entropy concept from information theory, in this paper we propose the generalized Shannon entropy function and R\'{e}nyi entropy function of the signal as the sparsity promoting regularizers. Both entropy functions are nonconvex, non-separable. Their local minimums only occur on the boundaries of the orthants in the Euclidean space. Compared to other popular objective functions, minimizing the generalized entropy functions adaptively promotes multiple high-energy coefficients while suppressing the rest low-energy coefficients. The corresponding optimization problems can be recasted into a series of reweighted $l_1$-norm minimization problems and then solved efficiently by adapting the FISTA. Sparse signal recovery experiments on both the simulated and real data show the proposed entropy functions minimization approaches perform better than other popular approaches and achieve state-of-the-art performances.

We present a collection of computer programs for the solution of linear and nonlinear semidefinite optimization problems. After briefly discussing the underlying algorithm, the generalized augmented Lagrangian method, we describe details of the specific programs for linear, bilinear and general nonlinear semidefinite optimization problems. For each of the programs we present typical application areas and examples.KeywordsPennonAugmented LagrangianCholesky MethodHessian-vector ProductsLinear Semidefinite Programming Problem (LSDP)These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

It is well know that the theory of minimal blocking sets is studied by several author. Another theory which is also studied by a large number of researchers is the theory of hyperplane arrangements. We can remark that the affine space AG(n, q) is the complement of the line at infinity in PG(n, q). Then AG(n, q) can be regarded as the complement of an hyperplane arrangement in PG(n, q)! Therefore the study of blocking sets in the affine space AG(n, q) is simply the study of blocking sets in the complement of a finite arrangement in PG(n, q). In this paper the author generalizes this remark starting to study the problem of existence of blocking sets in the complement of a given hyperplane arrangement in PG(n, q). As an example she solves the problem for the case of braid arrangement. Moreover she poses significant questions on this new and interesting problem.

Large, sparse, linear systems of equations arise frequently when constructing mathematical models of natural phenomena. Most often, these linear systems are fully constrained and can be solved via direct or iterative techniques. However, one important problem class requires solutions to underconstrained linear systems that maximize some objective function. These linear optimization problems are natural formulations of many business plans and often contain hundreds of equations with thousands of variables. Historically, linear optimization problems have been solved via the simplex method. Despite the excellent performance of the simplex method, the size of the optimization problems and the frequency of their solution make linear optimization a computationally taxing endeavor. This paper examines the performance of parallel variants of the simplex algorithm on the Intel iPSC, a message-based parallel system. Linear optimization test data are drawn from commercial sources and represent realistic problems. Analysis shows that the speedup obtained is sensitive to both the structure of the underlying data and the data partitioning.