Conic Reformulation Research Articles

Distributionally Robust Co-Optimization of Power Dispatch and Do-Not-Exceed Limits

To address the challenge of the renewable energy uncertainty, the ISO New England (ISO-NE) has proposed to apply do-not-exceed (DNE) limits, which represent the maximum nodal injection of renewable energy the grid can accommodate. Unfortunately, it appears challenging to compute DNE limits that simultaneously maintain the system flexibility and incorporate a large portion of the available renewable energy at the minimum cost. In addition, it is often challenging to accurately estimate the joint probability distribution of the renewable energy. In this paper, we propose a two-stage distributionally robust optimization model that co-optimizes the power dispatch and the DNE limits, by adopting an affinely adjustable power re-dispatch and an adjustable joint chance constraint that measures the renewable utilization. Notably, this model admits a second-order conic reformulation that can be efficiently solved by the commercial solvers (e.g., MOSEK). We conduct case studies based on modified IEEE test instances to demonstrate the effectiveness of the proposed approach and analyze the trade-off among the system flexibility, the renewable utilization, and the dispatch cost.

Nonconvex min\u2013max fractional quadratic problems under quadratic constraints: copositive relaxations

In this paper we address a min–max problem of fractional quadratic (not necessarily convex) over linear functions on a feasible set described by linear and (not necessarily convex) quadratic functions. We propose a conic reformulation on the cone of completely positive matrices. By relaxation, a doubly nonnegative conic formulation is used to provide lower bounds with evidence of very small gaps. It is known that in many solvers using Branch and Bound the optimal solution is obtained in early stages and a heavy computational price is paid in the next iterations to obtain the optimality certificate. To reduce this effort tight lower bounds are crucial. We will show empirical evidence that lower bounds provided by the copositive relaxation are able to substantially speed up a well known solver in obtaining the optimality certificate.

Distributionally Robust Chance Constrained Optimal Power Flow with Renewables: A Conic Reformulation

The uncertainty associated with renewable energy sources introduces significant challenges in optimal power flow (OPF) analysis. A variety of new approaches have been proposed that use chance constraints to limit line or bus overload risk in OPF models. Most existing formulations assume that the probability distributions associated with the uncertainty are known a priori or can be estimated accurately from empirical data, and/or use separate chance constraints for upper and lower line/bus limits. In this paper, we propose a data driven distributionally robust chance constrained optimal power flow model (DRCC-OPF), which ensures that the worst-case probability of violating both the upper and lower limit of a line/bus capacity under a wide family of distributions is small. Assuming that we can estimate the first and second moments of the underlying distributions based on empirical data, we propose an exact reformulation of DRCC-OPF as a tractable convex program. The key theoretical result behind this reformulation is a second-order cone programming (SOCP) reformulation of a general two-sided distributionally robust chance constrained set by lifting the set to a higher dimensional space. Our numerical study shows that the proposed SOCP formulation can be solved efficiently and that the results of our model are quite robust.

Improved Conic Reformulations for $K$-means Clustering

In this paper, we show that the popular $K$-means clustering problem can equivalently be reformulated as a conic program of polynomial size. The arising convex optimization problem is NP-hard, but ...

Conic approximation to quadratic optimization with linear complementarity constraints

This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity constraints.We provide a conic reformulation and its dual for the original problem such that these three problems share the same optimal objective value. Moreover, we show that the conic reformulation problem is attainable when the original problem has a nonempty and bounded feasible domain. Since the conic reformulation is in general a hard problem, some conic relaxations are further considered. We offer a condition under which both the semidefinite relaxation and its dual problem become strictly feasible for finding a lower bound in polynomial time. For more general cases, by adaptively refining the outer approximation of the feasible set, we propose a conic approximation algorithm to identify an optimal solution or an $$\epsilon $$∈-optimal solution of the original problem. A convergence proof is given under simple assumptions. Some computational results are included to illustrate the effectiveness of the proposed algorithm.

An integrated approach for airline scheduling, aircraft fleeting and routing with cruise speed control

To place an emphasis on profound relations among airline schedule planning problems and to mitigate the effect of unexpected delays, we integrate schedule design, fleet assignment and aircraft routing problems within a daily planning horizon while passengers’ connection service levels are ensured via chance constraints. We propose a nonlinear mixed integer programming model due to the nonlinear fuel consumption and CO2 emission cost terms in the objective function, which is handled by second order conic reformulation. The key contribution of this study is to take into account the cruise time control for the first time in an integrated model of these three stages of airline operations. Changing cruise times of flights in an integrated model enables to construct a schedule to increase utilization of fuel efficient aircraft and even to decrease total number of aircraft needed while satisfying the same service level and maintenance requirements for aircraft fleeting and routing. There is a critical tradeoff between the number of aircraft needed to fulfill the required flights and overall operational expenses. We also propose two heuristic methods to solve larger size problems. Finally, computational results using real data obtained from a major U.S. carrier are presented to demonstrate potential profitability in applying the proposed solution methods.

Minimizing the sum of a linear and a linear fractional function applying conic quadratic representation: continuous and discrete problems

This paper tries to minimize the sum of a linear and a linear fractional function over a closed convex set defined by some linear and conic quadratic constraints. At first, we represent some necessary and sufficient conditions for the pseudoconvexity of the problem. For each of the conditions, under some reasonable assumptions, an appropriate second-order cone programming (SOCP) reformulation of the problem is stated and a new applicable solution procedure is proposed. Efficiency of the proposed reformulations is demonstrated by numerical experiments. Secondly, we limit our attention to binary variables and derive a sufficient condition for SOCP representability. Using the experimental results on random instances, we show that the proposed conic reformulation is more efficient in comparison with the well-known linearization technique and it produces more eligible cuts for the branch and bound algorithm.

A distributionally robust perspective on uncertainty quantification and chance constrained programming

The objective of uncertainty quantification is to certify that a given physical, engineering or economic system satisfies multiple safety conditions with high probability. A more ambitious goal is to actively influence the system so as to guarantee and maintain its safety, a scenario which can be modeled through a chance constrained program. In this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry, unimodality or independence patterns. We delineate the watershed between tractability and intractability in ambiguity-averse uncertainty quantification and chance constrained programming. Using tools from distributionally robust optimization, we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones.

Conic approximation to nonconvex quadratic programming with convex quadratic constraints

In this paper, a conic reformulation and approximation is proposed for solving a nonconvex quadratic programming problem subject to several convex quadratic constraints. The original problem is transformed into a linear conic programming problem, which can be approximated by a sequence of linear conic programming problems over the dual cone of the cone of nonnegative quadratic functions. Since the dual cone of the cone of nonnegative quadratic functions has a linear matrix inequality representation, each linear conic programming problem in the sequence can be solved efficiently using the semidefinite programming techniques. In order to speed up the convergence of the approximation sequence and relieve the computational effort in solving the linear conic programming problems, an adaptive scheme is adopted in the proposed algorithm. We prove that the lower bounds generated by the linear conic programming problems converge to the optimal value of the original problem. Several numerical examples are used to illustrate how the algorithm works and the computational results demonstrate the efficiency of the proposed algorithm.

Capital requirements and optimal investment with solvency probability constraints

Quantifying economic capital and optimally allocating it into portfolios of financial instruments are two key topics in the asset–liability management of an insurance company. In general, these problems are studied in the literature by minimizing standard risk measures such as the value at risk and the conditional VaR. Motivated by Solvency II regulations, we introduce a novel optimization problem to solve for the optimal required capital and the portfolio structure simultaneously, when the ruin probability is used as an insurance solvency constraint. Besides the generic optimal required capital and portfolio problem formulation, we propose a two-model hierarchy of optimization models, where both models admit the so-called second-order conic reformulation, in turn making them particularly well suited for numerics. The first model, albeit naively asserting the normality of the returns on assets and liabilities, under minor further simplifications admits a closed-form solution—a set of formulas, which may be used as simple decision-making guidelines in the analysis of more complex scenarios. A potentially more realistic second model aims to represent the ‘heavy-tailed’ nature of an insurer's liabilities more accurately, while also allowing arbitrary distributions of asset returns via a semi-parametric approach. Extensive numerical simulations illustrate the sensitivity and robustness of the proposed approach relative to the model's parameters. In addition, we explore the potential of insurance risk diversification and discuss if combining several liabilities into a single insurance portfolio may always be beneficial for the insurer. Finally, we propose an extension of the model with an expected return on capital constraint added.

Quadratic optimization over one first-order cone

This paper studies the first-order cone constrained homogeneous quadratic programming problem. For efficient computation, the problem is reformulated as a linear conic programming problem. A union of second-order cones are designed to cover the first-order cone such that a sequence of linear conic programming problems can be constructed to approximate the conic reformulation. Since the cone of nonnegative quadratic forms over a union of second-order cones has a linear matrix inequalities representation, each linear conic programming problem in the sequence is polynomial-time solvable by applying semidefinite programming techniques. The convergence of the sequence is guaranteed when the union of second-order cones gets close enough to the first-order cone. In order to further improve the efficiency, an adaptive scheme is adopted. Numerical experiments are provided to illustrate the efficiency of the proposed approach.

A strong conic quadratic reformulation for machine-job assignment with controllable processing times

We describe a polynomial-size conic quadratic reformulation for a machine-job assignment problem with separable convex cost. Because the conic strengthening is based only on the objective of the problem, it can also be applied to other problems with similar cost functions. Computational results demonstrate the effectiveness of the conic reformulation.

Loss of Superlinear Convergence for an SQP-Type Method with Conic Constraints

In this short note we consider a sequential quadratic programming (SQP)--type method with conic subproblems and compare this method with a standard SQP method in which the conic constraint is linearized at each step. For both approaches we restrict our attention to convex subproblems since these are easy to solve and guarantee a certain global descent property. Using the example of a simple nonlinear program (NLP) and its conic reformulation we show that the SQP method with conic subproblems displays a slower rate of convergence than standard SQP methods. We then explain why an SQP subproblem that is based on a better approximation of the feasible set of the NLP results in a much slower algorithm.

?Cone-free? primal-dual path-following and potential-reduction polynomial time interior-point methods

We present a framework for designing and analyzing primal-dual interior-point methods for convex optimization. We assume that a self-concordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory and techniques of interior-point methods to the given good formulation of the problem (as is, without a conic reformulation) using the very usual primal central path concept and a less usual version of a dual path concept. We show that many of the advantages of the primal-dual interior-point techniques are available to us in this framework and therefore, they are not intrinsically tied to the conic reformulation and the logarithmic homogeneity of the underlying barrier function.

On self-concordant barrier functions for conic hulls and fractional programming

Given a self-concordant barrier function for a convex set[Figure not available: see fulltext.], we determine a self-concordant barrier function for the conic hull[Figure not available: see fulltext.] of[Figure not available: see fulltext.]. As our main result, we derive an "optimal" barrier for[Figure not available: see fulltext.] based on the barrier function for[Figure not available: see fulltext.]. Important applications of this result include the conic reformulation of a convex problem, and the solution of fractional programs by interior-point methods. The problem of minimizing a convex-concave fraction over some convex set can be solved by applying an interior-point method directly to the original nonconvex problem, or by applying an interior-point method to an equivalent convex reformulation of the original problem. Our main result allows to analyze the second approach showing that the rate of convergence is of the same order in both cases.