Piecewise Linear Valued CSPs Solvable by Linear Programming Relaxation

Submodular constraints play an important role both in theory and practice of valued constraint satisfaction problems (VCSPs). It has previously been shown, using results from the theory of combinatorial optimisation, that instances of VCSPs with submodular constraints can be minimised in polynomial time. However, the general algorithm is of order O(n 6) and hence rather impractical. In this paper, by using results from the theory of pseudo-Boolean optimisation, we identify several broad classes of submodular constraints over a Boolean domain which are expressible using binary submodular constraints, and hence can be minimised in cubic time. Furthermore, we describe how our results translate to certain optimisation problems arising in computer vision.

In this paper, a random search algorithm known as simulated annealing (SA) has been employed to optimize the sizing of a number of digital adder circuits. The SA algorithm is implemented in MATLAB; the cost function, a function of power and delay, is accurately computed using HSPICE for a 0.35 /spl mu/m technology. Using a piecewise linear and logarithmic cost function, the delay and power is optimized in an intelligent fashion. The results show a 60% reduction in power and a 65% reduction in delay with respect to previous designs based on analytical calculations.

On properties of linear programming model for acid rain abatement

Recently, increasing penetration of renewable energy generation has created challenges for power system operators to perform efficient power generation daily scheduling, due to the intermittent nature of the renewable generation and discrete decisions of each generation unit. Among all aspects to be considered, a unit commitment polytope is fundamental and embedded in the models at different stages of power system planning and operations. In this article, we focus on deriving polynomial-time algorithms for the unit commitment problems with a general convex cost function and piecewise linear cost function, respectively. We refine an time, where T represents the number of time periods, algorithm for the deterministic single-generator unit commitment problem with a general convex cost function and accordingly develop an extended formulation in a higher-dimensional space that can provide an integral solution, in which the physical meanings of the decision variables are described. This means the original problem can be solved as a convex program instead of a mixed-integer convex program. Furthermore, for the case in which the cost function is piecewise linear, by exploring the optimality conditions, we derive more efficient algorithms for both deterministic (i.e., time) and stochastic (i.e., time, where N represents the number of nodes in the stochastic scenario tree) single-generator unit commitment problems. We also develop the corresponding extended formulations for both deterministic and stochastic single-generator unit commitment problems that solve the original mixed-integer linear programs as linear programs. Similarly, physical meanings of the decision variables are explored to show the insights of the new modeling approach.

Globalization of design, manufacturing and logistics has made the production-transportation problem (PTP) an important aspect of the supply chain optimization. This problem is usually formulated with piecewise linear concave cost functions for both production and transportation costs, and then solved using a mixed-integer programming (MIP) algorithm. This paper studies the application of three different relaxations of the MIP formulation of the PTP and compares their computational efficiencies. A cutting-plane algorithm is adapted to improve the computational efficiency. Although the relaxations tend to provide similar optimal costs, they differ significantly with respect to computational complexity.

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than $2$ and $\omega(1)$, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than $1.5$ for the Vertex Cover Problem, and better than $2$ for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of $2-\varepsilon$ for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

An exact algorithm for two-dimensional vector packing problem with volumetric weight and general costs

This note gives a complete characterization of the cost function which is the dual of a homogeneous Cobb-Douglas production function.! The main result is that a continuous cost function has a first-order Taylor series approximation interpretation if and only if it is the dual of a homogeneous Cobb-Douglas production function. In other words, all Taylor series expansions of cost functions, when expanded jointly in prices and output, will collapse to the Cobb-Douglas form when terms of secondand higher-order are dropped. This is surprising, for it means not only that the cost function is homogeneous in prices as dictated by economic theory, but that in order to have a first-order Taylor series approximation interpretation it must be homogeneous in output as well. The analysis was motivated by the work of Barnett and others who have demonstrated that the second-order Taylor series expansions used to estimate cost, indirect utility, and profit functions frequently fail to satisfy certain regularity properties.2 The objective of this paper was to identify the functional forms of all continuous functions which provide a first-order Taylor series approximation to an arbitrary cost function at a point. While one would not expect to find any acceptable first-order approximations for cost functions since they cannot model arbitrary substitution possibilities, that expectation is rigorously confirmed here. The analysis is an extension of Eichhorn (1974, 1978) and Stehling (1975), and is similar to Fare and Sung (1986).

Submodular function minimization is a polynomially solvable combinatorial problem. Unfortunately the best known general-purpose algorithms have high-order polynomial time complexity. In many applications the objective function is locally defined in that it is the sum of cost functions (also known as soft or valued constraints) whose arities are bounded by a constant. We prove that every valued constraint satisfaction problem with submodular cost functions has an equivalent instance on the same constraint scopes in which the actual minimum value of the objective function is rendered explicit. Such an equivalent instance is the result of establishing optimal soft arc consistency and can hence be found by solving a linear program. From a practical point of view, this provides us with an alternative algorithm for minimizing locally defined submodular functions. From a theoretical point of view, this brings to light a previously unknown connection between submodularity and soft arc consistency.

We consider single processor preemptive scheduling with job-dependent setup times. In this model, a job-dependent setup time is incurred when a job is started for the first time, and each time it is restarted after preemption. This model is a common generalization of preemptive scheduling, and actually of non-preemptive scheduling as well. The objective is to minimize the sum of any general non-negative, non-decreasing cost functions of the completion times of the jobs -- this generalizes objectives of minimizing weighted flow time, flow-time squared, tardiness or the number of tardy jobs among many others. Our main result is a randomized polynomial time O(1)-speed O(1)-approximation algorithm for this problem. Without speedup, no polynomial time finite multiplicative approximation is possible unless P=NP. We extend the approach of Bansal et al. (FOCS 2007) of rounding a linear programming relaxation which accounts for costs incurred due to the non-preemptive nature of the schedule. A key new idea used in the rounding is that a point in the intersection polytope of two matroids can be decomposed as a convex combination of incidence vectors of sets that are independent in both matroids. In fact, we use this for the intersection of a partition matroid and a laminar matroid, in which case the decomposition can be found efficiently using network flows. Our approach gives a randomized polynomial time offline O(1)-speed O(1)-approximation algorithm for the broadcast scheduling problem with general cost functions as well.

We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible decomposition of monomial ideals. The gap can be computed in polynomial time when the dimension is fixed.

Intermittent energy sources demand temporal storages to balance generation and load, and batteries stand out as an alternative. However, the lifetime is limited, and cycling depth affects the battery degradation rate. Current stochastic multi-stage methods lack proper representation of battery degradation. This paper proposes a stochastic multi-stage model for optimizing battery operation in a microgrid considering battery degradation with a piece-wise linear cost function with uncertain wind power production and load. The model is solved using Stochastic Dual Dynamic Programming (SDDP) and is demonstrated on a 4-bus test case with limited import and export capacity to illustrate the battery degradation cost's impacts on the battery cycling strategy. The results show that the importance of a stochastic method is more pronounced when battery degradation is modelled.

When participating in electricity markets, owners of battery energy storage systems must bid in such a way that their revenues will at least cover their true cost of operation. Since cycle aging of battery cells represents a substantial part of this operating cost, the cost of battery degradation must be factored in these bids. However, existing models of battery degradation either do not fit market clearing software or do not reflect the actual battery aging mechanism. In this paper we model battery cycle aging using a piecewise linear cost function, an approach that provides a close approximation of the cycle aging mechanism of electrochemical batteries and can be incorporated easily into existing market dispatch programs. By defining the marginal aging cost of each battery cycle, we can assess the actual operating profitability of batteries. A case study demonstrates the effectiveness of the proposed model in maximizing the operating profit of a battery energy storage system taking part in the ISO New England energy and reserve markets.

We consider a quadratic programming (QP) problem (Π) of the form min x T C x subject to Ax ≥ b, x ≥ 0 where $${C\in {\mathbb R}^{n \times n}_+, {\rm rank}(C)=1}$$ and $${A\in {\mathbb R}^{m \times n}, b\in {\mathbb R}^m}$$ . We present an fully polynomial time approximation scheme (FPTAS) for this problem by reformulating the QP (Π) as a parameterized LP and “rounding” the optimal solution. Furthermore, our algorithm returns an extreme point solution of the polytope. Therefore, our results apply directly to 0–1 problems for which the convex hull of feasible integer solutions is known such as spanning tree, matchings and sub-modular flows. They also apply to problems for which the convex hull of the dominant of the feasible integer solutions is known such as s, t-shortest paths and s, t-min-cuts. For the above discrete problems, the quadratic program Π models the problem of obtaining an integer solution that minimizes the product of two linear non-negative cost functions.

One key challenge in creating a sustainable society is to make consumer demand adaptive to the supply of electricity. We propose a generalization for the multiagent coordination algorithm for partially-centralized demand side management. In this setting, a central unit buys the electricity for the whole group, while the individual agents make their own demand decisions, based on their private constraints and preferences. While the state-of-the-art algorithm achieves strong guarantees, i.e., efficiency, strict budget balance as well as weak incentive compatibility, the formulation requires a simple supply model with a 2-step increasing threshold price function. In this paper, we propose MDSM, a generalized formulation of the multiagent coordination algorithm and prove that it converges to the optimal solution for all convex piecewise linear cost functions. Further, we perform simulations based on real world consumption data to evaluate the scalability of the algorithm in respect to the number of thresholds. The results indicate that the convergence time of MDSM is not sensitive to the number of thresholds in the cost function. This means, the proposed algorithm could be used for a broader set of distributed scheduling problems.