Global optimisation of large-scale quadratic programs: application to short-term planning of industrial refinery-petrochemical complexes

This thesis is driven by an industrial problem arising in the short-term planning of an integrated refinery-petrochemical complex (IRPC) in Colombia. The IRPC of interest is composed of 60 industrial plants and a tank farm for crude mixing and fuel blending consisting of 30 additional units. It considers both domestic and imported crude oil supply, as well as refined product imports such as low sulphur diesel and alkylate. This gives rise to a large-scale mixed-integer quadratically constrained quadratic program (MIQCQP) comprising about 7,000 equality constraints with over 35,000 bilinear terms and 280 binary variables describing operating modes for the process units. Four realistic planning scenarios are recreated to study the performance of the algorithms developed through the thesis and compare them to commercial solvers. Local solvers such as SBB and DICOPT cannot reliably solve such large-scale MIQCQPs. Usually, it is challenging to even reach a feasible solution with these solvers, and a heuristic procedure is required to initialize the search. On the other hand, global solvers such as ANTIGONE and BARON determine a feasible solution for all the scenarios analysed, but they are unable to close the relaxation gap to less than 40% on average after 10h of CPU runtime. Overall, this industrial-size problem is thus intractable to global optimality in a monolithic way. The first main contribution of the thesis is a deterministic global optimisation algorithm based on cluster decomposition (CL) that divides the network into groups of process units according to their functionality. The algorithm runs through the sequences of clusters and proceeds by alternating between: (i) the (global) solution of a mixed-integer linear program (MILP), obtained by relaxing the bilinear terms based on their piecewise McCormick envelopes and a dynamic partition of their variable ranges, in order to determine an upper bound on the maximal profit; and (ii) the local solution of a quadratically-constrained quadratic program (QCQP), after fixing the binary variables and initializing the continuous variables to the relaxed MILP solution point, in order to determine a feasible solution (lower bound on the maximal profit). Applied to the base case scenario, the CL approach reaches a best solution of 2.964 MMUSD/day and a relaxation gap of 7.5%, a remarkable result for such challenging MIQCQP problem. The CL approach also vastly outperforms both ANTIGONE (2.634 MMUSD/day, 32% optimality gap) and BARON (2.687 MMUSD/day, 40% optimality gap). The second main contribution is a spatial Lagrangean decomposition, which entails decomposing the IRPC short-term planning problem into a collection of smaller subproblems that can be solved independently to determine an upper bound on the maximal profit. One advantage of this strategy is that each sub-problem can be solved to global optimality, potentially providing good initial points for the monolithic problem itself. It furthermore creates a virtual market for trading crude blends and intermediate refined–petrochemical streams and seeks an optimal trade-off in such a market, with the Lagrange multipliers acting as transfer prices. A decomposition over two to four is considered, which matches the crude management, refinery, petrochemical operations, and fuel blending sections of the IRPC. An optimality gap below 4% is achieved in all four scenarios considered, which is a significant improvement over the cluster decomposition algorithm.Open Acces

The Pareto-Frontier in a simple Mirrleesian model of income taxation

We characterize the Pareto-frontier in a simple Mirrleesian model of income taxation. We show how the second-best frontier which incorporates incentive constraints due to private information on productive abilities relates to the first-best frontier which takes only resource constraints into account. In particular, we argue that the second-best frontier can be interpreted as a Laer-curve. We also use this second-best frontier for a comparative statics analysis of how optimal income tax rates vary with the degree of inequity aversion, and for a characterization of optimal public-good provision. We show that a more inequity averse policy maker chooses tax schedules that are more redistributive and involve higher marginal tax rates, but chooses a lower public-goods provision level.Optimal Income Taxation, Public-good provision, Laer-Curve

Building Blocks in the Economics of Mandates

The paper constructs an asymmetric information model to investigate the efficiency and equity cases for government mandated benefits. A mandate can improve workers' insurance, and may also redistribute in favour of more "deserving" workers. The risk is that it may also reduce output. The more diverse are free market contracts – separating the various worker types – the more likely it is that such output effects will on balance serve to reduce welfare. It is shown that adverse effects can be reduced by restricting mandates to larger firms. An alternative to a mandate is direct government provision. We demonstrate that direct government provision has the advantage over mandates of preserving separations.asymmetric information, labour mandates, compensation packages

On the Desirability of Taxing Charitable Contributions

We develop a model that allows for public goods and status signaling through charitable contributions. This model provides a unified framework in which contributions are driven both by altruism and status signaling. We use this setup to re-examine the conventional practice of rendering a favorable tax treatment to charitable contributions.optimal taxation, re-distribution, charitable contributions, inequality

Mandated benefits, welfare, and heterogeneous firms

The paper constructs an asymmetric information model to investigate the efficiency and equity cases for government mandated benefits. A mandate can improve workers? insurance, and may also redistribute in favor of more "deserving" workers. The risk is that it may also reduce output. The more diverse are free market contracts - separating the various worker types - the more likely it is that such output effects will on balance serve to reduce welfare. It is shown that adverse effects can be mitigated by restricting mandates to "large" firms. An alternative to a mandate is direct government provision. We demonstrate that direct government provision may be superior to mandates by virtue of preserving separations. --

Optimal Income Taxation and Public Good Provision in a Two-Class Economy

This paper combines the problem of optimal income taxation with the free-rider problem in public good provision. There are two groups of individuals with private information on their earning ability and their valuation of a public good. Adjustments of the transfer system are needed to discourage the more productive from exaggerating the desirability of public good provision. Similarly, the less productive need to be prevented from understating their valuation. Relative to an optimal income tax, which focuses solely on earning ability, income transfers are increased whenever a public good is installed and are decreased otherwise

A Study of Lagrangean Decompositions and Dual Ascent Solvers for Graph Matching

We study the quadratic assignment problem, in computer vision also known as graph matching. Two leading solvers for this problem optimize the Lagrange decomposition duals with sub-gradient and dual ascent (also known as message passing) updates. We explore s direction further and propose several additional Lagrangean relaxations of the graph matching problem along with corresponding algorithms, which are all based on a common dual ascent framework. Our extensive empirical evaluation gives several theoretical insights and suggests a new state-of-the-art any-time solver for the considered problem. Our improvement over state-of-the-art is particularly visible on a new dataset with large-scale sparse problem instances containing more than 500 graph nodes each.Comment: Added acknowledgment

Rat Races and Glass Ceilings: Career Paths in Organizations.

In an ongoing organization, such as a large law parternship firm, employees are motivated not only by current rewards but also by the prospect of promotion, and the opportunity to influence policy and make the rules in the future. This leads to a dynamic programming problem in contract design. We model career design in such a firm as a recursive mechanism design problem in an overlapping generations environment.CONTRACTS ; GENERATIONS ; COSTS

Optimal Income Taxation and Public Good Provision in a Two-Class Economy

This paper combines the problem of optimal income taxation with the free-rider problem in public good provision. There are two groups of individuals with private information on their earning ability and their valuation of a public good. Adjustments of the transfer system are needed to discourage the more productive from exaggerating the desirability of public good provision. Similarly, the less productive need to be prevented from understating their valuation. Relative to an optimal income tax, which focuses solely on earning ability, income transfers are increased whenever a public good is installed and are decreased otherwise.Income Taxation; Public Good Provision; Revelation of Preferences; Two-dimensional Heterogeneity

A Lagrangean Relaxation and A Heuristic for the Pooling Problem

The pooling problem is one of the fundamental optimization problems encountered in the petroleum industry. In the pooling problem, final products are produced using two stages of blending operations. In the first stage, raw materials are mixed together to produce intermediate products. In the second stage, intermediate products and some of the raw materials are blended together according to product demand and quality requirements. Generally, the pooling problem is a nonlinear problem because the output stream qualities, which are unknown, depend on the volume, which is also unknown, and on the quality of the input streams. Specifically, nonlinearity and nonconvexity are due to the use of bilinear terms either in the quality constraints or in the objective function. Nonlinearity and nonconvexity result in several local optima, making the process of solving large-scale pooling problems to global optimality very challenging. Therefore, developing efficient heuristics for large-scale pooling problems is very desirable. Moreover, devising tight bounds on the global solutions is essential to assess the quality of the proposed heuristics. In this thesis, we use a Lagrangean relaxation approach where feasible solutions and lower bounds are generated for the pooling problem. The procedure targets all nonlinear constraints and penalizes their violation in the objective function. The resulting Lagrangean subproblem has a nonlinear objective function and linear constraints. The Lagrangean subproblem is reformulated as a mixed integer programming problem where the nonlinearities in the objective function are eliminated at the expense of using binary variables. The obtained Lagrangean lower bounds are strengthened using valid cuts that are based on the relaxed bilinear terms. In addition, at every iteration of the Lagrangean algorithm, the subproblem solutions are used to generate feasible solutions to the pooling problem. The procedure is applied to fifteen pooling problems collected from the literature. Some of these problems have a single quality and others have multiple qualities. Numerical results show that for eight solved cases, the obtained Lagrangean lower bounds are equal to the global optima, whereas for seven cases the obtained Lagrangean lower bound is on average 8.2% from the global optimum. Numerical results indicate the efficiency of the heuristic. For nine cases, the heuristic gives the global solution, and for the other cases the heuristic solutions are within 1.8% of the global optimum