Piecewise Linear Convex Function Research Articles

A Gradient-Aware Search Algorithm for Constrained Markov Decision Processes.

The canonical solution methodology for finite constrained Markov decision processes (CMDPs), where the objective is to maximize the expected infinite-horizon discounted rewards subject to the expected infinite-horizon discounted costs' constraints, is based on convex linear programming (LP). In this brief, we first prove that the optimization objective in the dual linear program of a finite CMDP is a piecewise linear convex (PWLC) function with respect to the Lagrange penalty multipliers. Next, we propose a novel, provably optimal, two-level gradient-aware search (GAS) algorithm which exploits the PWLC structure to find the optimal state-value function and Lagrange penalty multipliers of a finite CMDP. The proposed algorithm is applied in two stochastic control problems with constraints for performance comparison with binary search (BS), Lagrangian primal-dual optimization (PDO), and LP. Compared with the benchmark algorithms, it is shown that the proposed GAS algorithm converges to the optimal solution quickly without any hyperparameter tuning. In addition, the convergence speed of the proposed algorithm is not sensitive to the initialization of the Lagrange multipliers.

Sparse Bayesian inference with regularized Gaussian distributions *

Regularization is a common tool in variational inverse problems to impose assumptions on the parameters of the problem. One such assumption is sparsity, which is commonly promoted using lasso and total variation-like regularization. Although the solutions to many such regularized inverse problems can be considered as points of maximum probability of well-chosen posterior distributions, samples from these distributions are generally not sparse. In this paper, we present a framework for implicitly defining a probability distribution that combines the effects of sparsity imposing regularization with Gaussian distributions. Unlike continuous distributions, these implicit distributions can assign positive probability to sparse vectors. We study these regularized distributions for various regularization functions including total variation regularization and piecewise linear convex functions. We apply the developed theory to uncertainty quantification for Bayesian linear inverse problems and derive a Gibbs sampler for a Bayesian hierarchical model. To illustrate the difference between our sparsity-inducing framework and continuous distributions, we apply our framework to small-scale deblurring and computed tomography examples.

Scheduling a set of jobs with convex piecewise linear cost functions on a single-batch-processing machine

We investigate a general single-batch-processing machine scheduling problem of minimizing the total costs of all jobs with general convex piecewise linear cost functions, where the processing time, due dates, and sizes of jobs are non-uniform. Since the studied problem is NP-hard and the objective is irregular, we adopt a practical 3-step method that starts from a specific job sequence, finds the optimal and near-optimal schedules of the given job sequence, and then iteratively improves the schedules. For the restricted problem where the job sequence is given, we propose a dynamic programming algorithm to obtain the optimal schedule. To further reduce the running time, we also devise a span-limit tree search (SLTS) approach to find the near-optimal schedule. We then design a local search method to improve the solutions solved by SLTS. Finally, we design a branch and bound algorithm with a lower bound method for small-sized instances. All proposed methods are based on our analysis of the mathematical properties of the scheduling problem, and extensive numerical experiments are conducted to demonstrate their effectiveness and efficiency.

Just-in-Time Precast Production Scheduling Using Dominance Rule-Based Genetic Algorithm.

The objective of this study is to provide a model and method to assist fabricators in making appropriate precast production plans, coordinating factories prefabrication and on-site assembly in construction based on the just-in-time (JIT) philosophy. We propose a JIT precast production scheduling model for the precast production of steel box girders in the Hong Kong-Zhuhai-Macau (HZM) bridge construction project. In order to minimize the total early/tardy costs for this JIT scheduling model, we first present a job batching algorithm (JBA) to group series of jobs into several batches. Then, we model the batch cost function as a piecewise linear convex function and derive its optimal solution, which guides us to propose an optimal shifting algorithm (OSA) to locate the optimal starting time of each batch and minimize the batch cost. In order to get the best job sequence for minimizing the total early/tardy costs and improve search efficiency, we introduce a dominance rule for early/tardy scheduling problem and propose a dominance rule-based genetic algorithm (DBGA) embedded with an optimal timing algorithm, which can find the best job sequence as well as the corresponding optimal schedule. The real-world case study of HZM bridge project demonstrates that our proposed model and algorithm can assist construction practitioners to make better decision on precast production scheduling compared to empirical rule in current engineering practice. In addition, numerical studies prove that our proposed algorithm has a better performance on both effectiveness and efficiency due to the dominance rule.

Computational Experiments for the r(α)-Algorithm with an Accelerated Implementation of Space Dilation

In the article, a modification of the r(α)-algorithm with adaptive line search which implements accelerated space dilation operation is considered. In classical r(α)-algorithm with adaptive line search, space is dilated along the difference of the two successive subgradients. In its modified variant, space is dilated along the direction which is determined by the largest absolute value components of the vector of difference of the two successive subgradients. The number of such components varies at each iteration of the algorithm and depends on a control parameter. Thus, at each iteration only corresponding subspace of a smaller dimension is dilated and the number of multiplication operations used for this decreases from 2n2 + 3n to 2nm + 2m + n, where n is the dimension of the space, m is the number of non-zero components of the space dilation direction. Description of the modified r(α)-algorithm with adaptive line search which implements accelerated space dilation operation is given. Results of the numerical experiments of minimization of the two classes of convex piecewise linear functions and quadratic strongly convex functions are presented. To solve the problems C++ implementations of the classical r(α)-algorithm with an adaptive line search ralgb5a and its modified version with accelerated space dilation ralgb5at were used. Obtained results show that for minimization of the piecewise linear functions compared with the classical r(α)-algorithm its modified version requires approximately the same number of iterations to converge but uses a significantly smaller number of multiplication operations. For minimization of the quadratic strongly convex functions modified r(α)-algorithm requires a smaller number of iterations to converge and as a result has a smaller execution time. Therefore, it is concluded that the variants of the r-algorithm that implements accelerated space dilation operation along directions which approximate the difference of the two successive subgradients may be quite promising since for some classes of functions they might better take into account the shape of the function’s ravines and with appropriate software implementation have a smaller execution time. Keywords: space dilation operator, r(α)-algorithm, piecewise linear function.

GNU Octave and Python Implementation of Shor's r-Algorithm with Adaptive Step Control

r-algorithms, or subgradient methods with dilation of space in the direction of the difference of two sequential subgradients, were proposed by N.Z.Shor in 1970 in his doctoral thesis. Respective software implementations proved to be competitive with the most effective methods for smooth ill-conditioned problems, both in terms of reliability and calculation time and accuracy of results. The article is devoted to the description of two software implementations of Shor's r-algorithm modification with a constant coefficient of space dilation and adaptive step control. The first program is implemented using the GNU Octave, and the second program is implemented using Python. Material of the paper is presented in three sections. In the first section, we describe a modification of the r-algorithm with a constant coefficient of space dilation in the direction of the difference of two sequential subgradients and an adaptive method for step size adjustment in the direction of the antisubgradient in the transformed space of variables. The software implementation of this modification is presented in the form of octave-function ralgb5a, which allows to find either approximation of the minimum point of a convex function, or approximation of the maximum point of the concave function. The code of the ralgb5a function is given with a brief description of its input and output parameters. The second section describes test experiments to investigate efficiency of the octave-function ralgb5a to maximizing the piecewise linear concave function, which is obtained using the method of non-smooth penalty functions for the linear programming problem. Another example represents minimization of the piecewise linear convex function, which corresponds to the method of least modules. Results of these computational experiments for test problems with 200, 500, 1000, 1500 and 2000 variables are presented to demonstrate the effective operation of the octave-function ralgb5a. The third section describes the python function ralgb5a and provides its code with a description of the input and output parameters. It is show, how the ralgb5a function can be accelerated by setting two parameters. The results of computational experiments to solve the test problem using the method of least modules for 5,000 variables and 10,000 observations are presented. Keywords: r-algorithm, linear programming problem, nonsmooth penalty function, least modulus method, GNU Octave, python.

Random Tessellations and Gibbsian Solutions of Hamilton–Jacobi Equations

We pursue two goals in this article. As our first goal, we construct a family $\mathcal{M}_G$ of Gibbs like measures on the set of piecewise linear convex functions $g:\mathbb{R}^2\to\mathbb{R}$. It turns out that there is a one-to-one correspondence between the gradient of such convex functions and $\textit{Laguerre tessellations}$. Each cell in a Laguerre tessellation is a convex polygon that is marked by a vector $\rho\in\mathbb{R}^2$. Each measure $\nu^f\in\mathcal{M}_G$ in our family is uniquely characterized by a kernel $f(x,\rho^-,\rho^+)$, which represents the rate at which a line separating two cells associated with marks $\rho^-$ and $\rho^+$ passes through $x$. To construct our measures, we give a precise recipe for the law of the restriction of our tessellation to a box. This recipe involves a boundary condition, and a dynamical description of our random tessellation inside the box. As we enlarge the box, the consistency of these random tessellations requires that the kernel satisfies a suitable kinetic like PDE. As our second goal, we study the invariance of the set $\mathcal{M}_G$ with respect to the dynamics of such Hamilton-Jacobi PDEs. In particular we $\textit{conjecture}$ the invariance of a suitable subfamily $\widehat{\mathcal{M}_G}$ of $\mathcal{M}_G$. More precisely, we expect that if the initial slope $u_x(\cdot,0)$ is selected according to a measure $\nu^{f}\in \widehat{\mathcal{M}_G}$, then at a later time the law of $u_x(\cdot, t)$ is given by a measure $\nu^{\Theta_t(f)}\in\widehat{\mathcal{M}_G}$, for a suitable kernel $\Theta_t(f)$. As we vary $t$, the kernel $\Theta_t(f)$ must satisfy a suitable kinetic equation. We remark that the function $u$ is also piecewise linear convex function in $(x,t)$, and its law is an example of a Gibbs-like measure on the set of Laguerre tessellations of certain convex subsets of $\mathbb{R}^3$.

Interior Point Methods Can Exploit Structure of Convex Piecewise Linear Functions with Application in Radiation Therapy

Auxiliary variables are often used to model a convex piecewise linear function in the framework of linear optimization. This work shows that such variables yield a block diagonal plus low rank structure in the reduced KKT system of the dual problem. We show how the structure can be detected efficiently and derive the linear algebra formulas for an interior point method which exploits such a structure. The structure is detected in 36% of the cases in Netlib. Numerical results on the inverse planning problem in radiation therapy show an order of magnitude speed-up compared to the state-of-the-art interior point solver CPLEX and considerable improvements in dose distribution compared to current algorithms.

An extended formulation for two-stage stochastic unit commitment with reserves

We derive extended formulations for the two-stage stochastic unit commitment with reserves. Our approach can capture most physical characteristics of a complicated thermal generator including min-up/-down time limit, time-dependent start-up costs, start-up/shut-down ramping, general ramping in the stable region, and reserve amount commitments. The extended formulations include both the convex hull descriptions for the feasible region and convex envelopes for the convex piece-wise linear and convex quadratic objective functions, which provide the optimal objective value for the original problem.

Weapon-target assignment problem: exact and approximate solution algorithms

The Weapon-Target Assignment (WTA) problem aims to assign a set of weapons to a number of assets (targets), such that the expected value of survived targets is minimized. The WTA problem is a nonlinear combinatorial optimization problem known to be NP-hard. This paper applies several existing techniques to linearize the WTA problem. One linearization technique (Camm et al. in Oper Res 50(6):946–955, 2002) approximates the nonlinear terms of the WTA problem via convex piecewise linear functions and provides heuristic solutions to the WTA problem. Such approximation problems are, though, relatively easy to solve from the computational point of view even for large-scale problem instances. Another approach proposed by O’Hanley et al. (Eur J Oper Res 230(1):63–75, 2013) linearizes the WTA problem exactly at the expense of incorporating a significant number of additional variables and constraints, which makes many large-scale problem instances intractable. Motivated by the results of computational experiments with these existing solution approaches, a specialized new exact solution approach is developed, which is called branch-and-adjust. The proposed solution approach involves the compact piecewise linear convex under-approximation of the WTA objective function and solves the WTA problem exactly. The algorithm builds on top of any existing branch-and-cut or branch-and-bound algorithm and can be implemented using the tools provided by state-of-the-art mixed integer linear programming solvers. Numerical experiments demonstrate that the proposed specialized algorithm is capable of handling very large scale problem instances with up to 1500 weapons and 1000 targets, obtaining solutions with optimality gaps of up to \(2.0\%\) within 2 h of computational runtime.

Convex minorants and the fluctuation theory of Lévy processes

We establish a novel characterisation of the law of the convex minorant of any L\'evy process. Our self-contained elementary proof is based on the analysis of piecewise linear convex functions and requires only very basic properties of L\'evy processes. Our main result provides a new simple and self-contained approach to the fluctuation theory of L\'evy processes, circumventing local time and excursion theory. Easy corollaries include classical theorems, such as Rogozin's regularity criterion, Spitzer's identities and the Wiener-Hopf factorisation, as well as a novel factorisation identity.

PL DENSITY INVARIANT FOR TYPE II DEGENERATING K3 SURFACES, MODULI COMPACTIFICATION AND HYPER-KÄHLER METRIC

AbstractA protagonist here is a new-type invariant for type II degenerations of K3 surfaces, which is explicit piecewise linear convex function from the interval with at most $18$ nonlinear points. Forgetting its actual function behavior, it also classifies the type II degenerations into several combinatorial types, depending on the type of root lattices as appeared in classical examples.From differential geometric viewpoint, the function is obtained as the density function of the limit measure on the collapsing hyper-Kähler metrics to conjectural segments, as in [HSZ19]. On the way, we also reconstruct a moduli compactification of elliptic K3 surfaces by [AB19], [ABE20], [Brun15] in a more elementary manner, and analyze the cusps more explicitly.We also interpret the glued hyper-Kähler fibration of [HSVZ18] as a special case from our viewpoint, and discuss other cases, and possible relations with Landau–Ginzburg models in the mirror symmetry context.

Use of the Shor’s r-Algorithm in Linear Robust Optimization Problems

The paper is devoted to the description of a new approach to the construction of algorithms for solving linear programming problems (LP-problems), in which the number of constraints is much greater than the number of variables. It is based on the use of a modification of the r-algorithm to solve the problem of minimizing a nonsmooth function, which is equivalent to LP problem. The advantages of the approach are demonstrated on the linear robust optimization problem and the robust parameters estimation problem using the least moduli method. The developed octave programs are designed to solve LP problems with a very large number of constraints, for which the use of standard software from linear programming is either impossible or impractical, because it requires significant computing resources. The material of the paper is presented in three sections. In the first section for the problem of minimizing a convex function we describe a modification of the r-algorithm with a constant coefficient of space dilation in the direction of the difference of two successive subgradients and an adaptive method for step size adjustment in the direction of the antisubgradient in the transformed space of variables. The software implementation of this modification is presented in the form of Octave function ralgb5a, which allows to find or approximation of the minimum point of a convex function, or approximation of the maximum point of the concave function. The code of the ralgb5a function is given with a brief description of its input and output parameters. In the second section, a method for solving the LP problem is presented using a nonsmooth penalty function in the form of maximum function and the construction of an auxiliary problem of unconstrained minimization of a convex piecewise linear function. The choice of the finite penalty coefficient ensures equivalence between the LP-problem and the auxiliary problem, and the latter is solved using the ralgb5a program. The results of computational experiments in GNU Octave for solving test LP-problems with the number of constraints from two hundred thousand to fifty million and the number of variables from ten to fifty are presented. The third section presents least moduli method that is robust to abnormal observations or "outliers". The method uses the problem of unconstrained minimization of a convex piecewise linear function, and is solved using the ralgb5a program. The results of computational experiments in GNU Octave for solving test problems with a large number of observations (from two hundred thousand to five million) and a small number of unknown parameters (from ten to one hundred) are presented. They demonstrate the superiority of the developed programs over well-known linear programming software such as the GLPK package. Keywords: robust optimization, linear programming problem, nonsmooth penalty function, r-algorithm, least modulus method, GNU Octave.

Stochastic economic dispatch of power system with multiple wind farms and pumped‐storage hydro stations using approximate dynamic programming

The stochastic economic dispatch problem of power system with multiple wind farms and pumped-storage hydro stations is formulated as a specific stochastic dynamic programming (DP) model, i.e. stochastic storage model, it is impossible to obtain an accurate solution due to the curse of dimensionality. Based on the approximate DP (ADP) method, the stochastic storage model can be transformed into a series of mixed-integer linear programming (MILP) models by describing the approximate value functions (AVFs) as convex piecewise linear functions in post-decision states. The AVFs are first initialised using the results of the deterministic model under a forecast scenario of wind farm output and then trained by scanning stochastic sampling scenarios consecutively with the successive projective approximation routine algorithm. To obtain a near-optimal day-ahead dispatch scheme, the forecast scenario is substituted into the MILP models expressed by the trained AVFs and is solved forward through each time interval. The network constraints are incorporated by the while-loop detection of critical lines. Test results on an actual provincial power system and the modified IEEE 39-bus system, including the comparison among the ADP, DP, scenario-based and chance-constrained programming methods, demonstrate the feasibility and efficiency of the proposed model and algorithm.

Mixed integer programming with convex/concave constraints: Fixed-parameter tractability and applications to multicovering and voting

A classic result of Lenstra [Math. Oper. Res. 1983] says that an integer linear program can be solved in fixed-parameter tractable (FPT) time for the parameterization by the number of variables. We extend this result by incorporating piecewise linear convex or concave functions to our (mixed) integer programs. This general technique allows us to analyze the parameterized complexity of a number of classic NP-hard computational problems. In particular, we prove that Weighted Set Multicover is in FPT when parameterized by the number of elements to cover, and that there exists an FPT-time approximation scheme for Multiset Multicover for the same parameter—this is our most technical result. Further, we use our general technique to prove that a number of problems from computational social choice (e.g., problems related to bribery and control in elections) are in FPT when parameterized by the number of candidates. For bribery, this resolves a nearly 10-year old family of open problems, and for weighted electoral control of Approval voting, this improves some previously known XP-memberships to FPT-memberships.

Impact of Energy Storage on Economic Dispatch of Distribution Systems: A Multi-Parametric Linear Programming Approach and its Implications

As a main flexible resource, energy storage helps smooth the volatility of renewable generation and reshape the load profile. This paper aims to characterize the impact of energy storage unit on the economic operation of distribution systems in a geometric manner that is convenient for visualization. Posed as a multi-parametric linear programming problem, the optimal operation cost is explicitly expressed as a convex piecewise linear function in the MW/MWh parameter of the energy storage unit. Based on duality theory, a dual linear programming based algorithm is proposed to calculate an approximate optimal value function (OVF) and critical regions, circumventing the difficulty of degeneracy, a common challenge in the existing multi-parametric linear programming solvers. When the uncertainty of renewable generation is considered, the expected OVF can be readily established based on OVFs in the individual scenarios, which is scalable in the number of scenarios. The OVF delivers abundant sensitivity information that is useful in energy storage sizing. Leveraging the OVFs, a robust stochastic optimization model is proposed to determine the optimal MW-MWh size of the storage unit subject to a given budget, which gives rise to a simple linear program. Case study provides a clear sketch of the outcome of the proposed method, and suggests that the optimal energy-power ratio of an energy storage unit is between 5 and 6 from the economical perspective.

Radio Communications Interdiction Problem under deterministic and probabilistic jamming

The Radio Communications Interdiction Problem (RCIP) seeks to identify the locations of transmitters on the battlefield that will lead to a robust radio communications network by anticipating the effects of intentional radio jamming attacks used by an adversary during electronic warfare. RCIP is a sequential game defined between two opponents that target each other’s military units in a conventional warfare. First, a defender locates a limited number of transmitters on the defender’s side of the battlefield to optimize the relay of information among its units. After observing the locations of radio transmitters, an attacker locates a limited number of radio jammers on the attacker’s side to disrupt the communication network of the defender. We formulate RCIP as a binary bilevel (max–min) programming problem, present the equivalent single level formulation, and propose an exact solution method using a decomposition scheme. We enhance the performance of the algorithm by utilizing dominance relations, preprocessing, and initial starting heuristics. To reflect a more realistic jamming representation, we also introduce the probabilistic version of RCIP where a jamming probability is associated at each receiver site as a function of the prevalent jamming to signal ratios leading to an expected coverage of receivers as an objective function. We approximate the nonlinearity in the jamming probability function using a piecewise linear convex function and solve this version by adapting the decomposition algorithm constructed for RCIP. Our extensive computational results on realistic scenarios show the efficacy of the solution approaches and provide valuable tactical insights.

Complexity Estimation for an Algorithm of Searching for Zero of a Piecewise Linear Convex Function

It is known that the problem of the orthogonal projection of a point to the standard simplex can be reduced to solution of a scalar equation. In this article, the complexity is analyzed of an algorithm of searching for zero of a piecewise linear convex function which is proposed in [30]. The analysis is carried out of the best and worst cases of the input data for the algorithm. To this end, the largest and smallest numbers of iterations of the algorithm are studied as functions of the size of the input data. It is shown that, in the case of equality of elements of the input set, the algorithm performs the smallest number of iterations. In the case of different elements of the input set, the number of iterations is maximal and depends rather weakly on the particular values of the elements of the set. The results of numerical experiments with random input data of large dimension are presented.

The inverse 1-center problem on cycles with variable edge lengths

We consider the problem of modifying edge lengths of a cycle at minimum total costs so as to make a prespecified vertex an (absolute) 1-center of the cycle with respect to the new edge legths. We call this problem the inverse 1-center problem on a cycle. To solve this problem, we first construct the so-called optimality criterion for a vertex to be a 1-center. Based on the optimality criterion, it is shown that the problem can be separated into linearly many subproblems. For a predetermined subproblem, we apply a parameterization approach to formulate it as a minimization problem of a piecewise linear convex function with a connected feasible region. Hence, it is shown that the problem can be solved in $$O(n^2 \log n)$$ time, where n is the number of vertices in the cycle.

Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form

We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form $$A(x) : D^2 u(x) = f(x)$$ in a bounded but not necessarily convex domain $$\Omega $$ and study it in the max norm. The fine scale is given by the meshsize h, whereas the coarse scale $$\epsilon $$ is dictated by an integro-differential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle for any uniformly positive definite matrix A provided that the mesh is face weakly acute. We establish a discrete Alexandroff–Bakelman–Pucci (ABP) estimate which is suitable for finite element analysis. Its proof relies on a discrete Alexandroff estimate which expresses the min of a convex piecewise linear function in terms of the measure of its sub-differential, and thus of jumps of its gradient. The discrete ABP estimate leads, under suitable regularity assumptions on A and u, to pointwise error estimates of the form $$\begin{aligned} \Vert \,u - u^{\epsilon }_h\,\Vert _{L^{\infty }(\Omega )} \le \, C(A,u) \, h^{2\alpha /(2 + \alpha )} \big | \ln h \big | \qquad 0< \alpha \le 2, \end{aligned}$$ provided $$\epsilon \approx h^{2/(2+\alpha )}$$ . Such a convergence rate is at best of order $$ h \big | \ln h \big |$$ , which turns out to be quasi-optimal.