Bregman primal–dual first-order method and application to sparse semidefinite programming

We present a new variant of the Chambolle–Pock primal–dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method is a line search procedure for selecting suitable step sizes. The line search obviates the need for estimating the norm of the constraint matrix and the strong convexity constant of the Bregman kernel. As an application, we discuss the centering problem in large-scale semidefinite programming with sparse coefficient matrices. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as the distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.

Online Linear Programming: Dual Convergence, New Algorithms, and Regret Bounds

We study an online linear programming (OLP) problem under a random input model in which the columns of the constraint matrix along with the corresponding coefficients in the objective function are independently and identically drawn from an unknown distribution and revealed sequentially over time. Virtually all existing online algorithms were based on learning the dual optimal solutions/prices of the linear programs (LPs), and their analyses were focused on the aggregate objective value and solving the packing LP, where all coefficients in the constraint matrix and objective are nonnegative. However, two major open questions were as follows. (i) Does the set of LP optimal dual prices learned in the existing algorithms converge to those of the “offline” LP? (ii) Could the results be extended to general LP problems where the coefficients can be either positive or negative? We resolve these two questions by establishing convergence results for the dual prices under moderate regularity conditions for general LP problems. Specifically, we identify an equivalent form of the dual problem that relates the dual LP with a sample average approximation to a stochastic program. Furthermore, we propose a new type of OLP algorithm, action-history-dependent learning algorithm, which improves the previous algorithm performances by taking into account the past input data and the past decisions/actions. We derive an [Formula: see text] regret bound (under a locally strong convexity and smoothness condition) for the proposed algorithm, against the [Formula: see text] bound for typical dual-price learning algorithms, where n is the number of decision variables. Numerical experiments demonstrate the effectiveness of the proposed algorithm and the action-history-dependent design.

Application of Digital Mining Facing Information Fusion Technology in the Field of National Costume Culture Design

In order to improve the design effect of minority clothing, according to the needs of minority clothing design, this paper uses data mining and Internet of Things technologies to construct an intelligent ethnic clothing design system and builds an intelligent clothing design system that meets customer needs based on the idea of human-computer interaction. In data processing, this paper uses the constraint spectrum clustering algorithm to take the Laplacian matrix and the constraint matrix as input and finally outputs a clustering indicator vector to improve the data processing effect of minority clothing design. Finally, this paper verifies the performance of the system designed in this paper through experiments. From the experimental research, it can be known that the minority clothing design system based on the Internet of Things and data mining constructed in this paper has a certain effect and can effectively improve the minority clothing design effect.

Two-stage spline-approximation in linear structure routing

In the article, computer design of routes of linear structures is considered as a spline approximation problem. A fundamental feature of the corresponding design tasks is that the plan and longitudinal profile of the route consist of elements of a given type. Depending on the type of linear structure, line segments, arcs of circles, parabolas of the second degree, clothoids, etc. are used. In any case, the design result is a curve consisting of the required sequence of elements of a given type. At the points of conjugation, the elements have a common tangent, and in the most difficult case, a common curvature. Such curves are usually called splines. In contrast to other applications of splines in the design of routes of linear structures, it is necessary to take into account numerous restrictions on the parameters of spline elements arising from the need to comply with technical standards in order to ensure the normal operation of the future structure. Technical constraints are formalized as a system of inequalities. The main distinguishing feature of the considered design problems is that the number of elements of the required spline is usually unknown and must be determined in the process of solving the problem. This circumstance fundamentally complicates the problem and does not allow using mathematical models and nonlinear programming algorithms to solve it, since the dimension of the problem is unknown. The article proposes a two-stage scheme for spline approximation of a plane curve. The curve is given by a sequence of points, and the number of spline elements is unknown. At the first stage, the number of spline elements and an approximate solution to the approximation problem are determined. The method of dynamic programming with minimization of the sum of squares of deviations at the initial points is used. At the second stage, the parameters of the spline element are optimized. The algorithms of nonlinear programming are used. They were developed taking into account the peculiarities of the system of constraints. Moreover, at each iteration of the optimization process for the corresponding set of active constraints, a basis is constructed in the null space of the constraint matrix and in the subspace – its complement. This makes it possible to find the direction of descent and solve the problem of excluding constraints from the active set without solving systems of linear equations. As an objective function, along with the traditionally used sum of squares of the deviations of the initial points from the spline, the article proposes other functions taking into account the specificity of a particular project task.

Landing gear disassembly sequence planning using multi-level constraint matrix ant colony algorithm

The life cycle of most complex engineering systems is greatly a function of maintenance. Generally, most maintenance operation usually requires the removal of failed part. Disassembly sequence planning is an optimization program that seeks to identify the optimal sequence for the removal of the failed part. Most studies in this area usually, use single constraint matrix while implementing varied complex algorithm to identify the optimal sequence that saves time associated with carrying out maintenance operation. The used of single constraint matrix typically has the drawback of computer higher storage requirement as well as time consumption. To address this problem, this study proposes Multi-Level Constraint Matrix Ant Colony Algorithm (MLCMACA). MLCMACA efficiency was validated using complex aircraft landing gear systems in comparison with genetic algorithms. The result shows MLCMACA superior performance from the perspective of reduced search time and faster tracking of optimal disassembly sequence. Hence is recommended for handling of disassembly sequence planning problems.

Improved particle swarm optimization for mean-variance-Yager entropy-social responsibility portfolio with complex reality constraints

PurposeThe weighted evaluation function method with normalized objective functions is used to transform the proposed multi-objective model into a single objective one, which reflects the investors' preference for returns, risks and social responsibility by adjusting the weights. Finally, an example is given to illustrate the solution steps of the model and the effectiveness of the algorithm.Design/methodology/approachBased on the possibility theory, assuming that the future returns of each asset are trapezoidal fuzzy numbers, a mean-variance-Yager entropy-social responsibility model is constructed including piecewise linear transaction costs and risk-free assets. The model proposed in this paper includes six constraints, the investment proportion sum, the non-negativity proportion, the ceiling and floor, the pre-assignment, the cardinality and the round lot constraints. In addition, considering the special round lot constraint, the proposed model is transformed into an integer programming problem.FindingsThe effects of different constraints and transaction costs on the effective frontier of the portfolio are analyzed, which not only assists investors to make decisions close to their expectations by setting appropriate parameters but also provides constructive suggestions through the overall performance of each asset.Originality/valueThere are two improvements in the improved particle swarm optimization algorithm: one is that the complex constraints are specifically satisfied by using a renewable 0–1 random constraint matrix and random scaling factors instead of fixed ones; the other is eliminating the particles with poor fitness and randomly adding some new particles that satisfy all the constraints to achieve the goal of global search as much as possible.

About the complexity of two-stage stochastic IPs

We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form $$\max \{ c^T x \mid {\mathcal {A}} x = b, \,l \le x \le u,\, x \in {\mathbb {Z}}^{s + nt} \}$$ max { c T x ∣ A x = b , l ≤ x ≤ u , x ∈ Z s + n t } where the constraint matrix $${\mathcal {A}} \in {\mathbb {Z}}^{r n \times s +nt}$$ A ∈ Z r n × s + n t consists roughly of n repetitions of a matrix $$A \in {\mathbb {Z}}^{r \times s}$$ A ∈ Z r × s on the vertical line and n repetitions of a matrix $$B \in {\mathbb {Z}}^{r \times t}$$ B ∈ Z r × t on the diagonal. In this paper we improve upon an algorithmic result by Hemmecke and Schultz from 2003 [Hemmecke and Schultz, Math. Prog. 2003] to solve 2-stage stochastic IPs. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters r, s, t and $$\Delta $$ Δ , where $$\Delta $$ Δ is the largest entry of the constraint matrix. Our new improved bound however is obtained by a novel theorem which argues about intersections of paths in a vector space. As a result of our new bound we obtain an algorithm to solve 2-stage stochastic IPs in time $$f(r,s,\Delta ) \cdot \mathrm {poly}(n,t)$$ f ( r , s , Δ ) · poly ( n , t ) , where f is a doubly exponential function. To complement our result, we also prove a doubly exponential lower bound for the size of the augmenting steps.

Watt Six-Bar Compliant Mechanism Analysis based on Kinematic and Dynamic Responses

In this study, a complete guide to kinematic and kinetic analyses of a Watt type six-bar compliant mechanism is conducted incorporating the flexible buckling of the initially straight element. In the analysis procedure, the hybrid utilization of the pseudo-rigid-body model (PRBM) and the nonlinear Elastic theory of beam buckling is presented. This partially compliant mechanism comprises three rigid links and two flexible links. The kinematic analyses of the mechanisms are done by using the vector loop closure equations, the PRBM of a large deflection cantilever beam, and derivation of nonlinear algebraic equations considering the quasi-static equilibrium and load-deflection curve of the flexible parts. Each of the elastic parts makes up a buckling pinned-pinned flexible Euler beam. The vector loop equations are combined with Newton-Euler dynamic formulations to provide the simultaneous constraint matrix. After these operations, the full mechanism is simulated to get both accelerations and forces for each time step. Finally, the design method is validated through experimental results. The findings derived from the combination of buckling Elastica solution and PRBM approach enable the analysis of Watt's six-bar compliant mechanism.