Strong Bounds for Linear Programs with Cardinality Limited Violation (CLV) Constraint Systems Ronald L. Rardin University of Arkansas Mark Langer, M.D. Indiana University Ali Tuncel Purdue University Jean-Philippe Richard Purdue University Workshop on Mixed Integer Programming August 4-7, 2008

Models LPCLV have all constraints and the objective linear, but some constraints belong to systems where up to k of m may be violated LP’s with CLV Constraints

If some outer limit on violation is available, the CLV system is easily modeled But a difficult MILP often results, especially when there is a great difference between and MIP Formulation of CLV Constraints

LPCLV models arise in Value at Risk portfolio optimization, where decision variables x j represent the investment from budget P in asset j, and CLV constraints reflect risk under various scenarios i, at most k of which may exceed b Application: Portfolio Optimization

Application: Radiation Therapy (Fluence) Planning Radiation is delivered by an accelerator that rotates 360 degrees around the patient Full beam area is large, around 10 cm square. However, Intensity Modulated Radiation Therapy (IMRT) can deliver shaped intensity maps composed of 3-10mm rectangular beamlets with independent fluence

Application: Radiation Therapy (Fluence) Planning The goal of intensity/fluence/timeon planning optimization is to assign optimal beamlet intensities x j so that prescribed doses are delivered to different types of tissues: Primary Target: Tumor region. Maximize dose within homogeneity requirements Secondary Targets: Minimum dose limits to eliminate risk of microscopic infection. Healthy/Normal Tissues: Maximum dose limits on all or specified fractions of surrounding tissues to avoid complications. Dose-volume limits are ones on fractions of normal tissue volume (e.g. no more than 20% of points in the tissue can receive a dose exceeding 60)

Tissues are modeled as collections of imbedded points i Dose at any point is approximately linear in beamlet fluence x j Leads to a CLV system for each dose-volume constrained tissue with other constraints linear Dose-Volume Tissues

Computational Challenge – Hard Cases The difference between b and  b values influences the initial optimality gap.

Rounding Heuristic An LP-rounding heuristic often gives feasible integer solutions 1. Solve the LP relaxation 2. Sort the points in each dose-volume tissue by LP relaxation dose 3. Force all but the highest-dose k to satisfy b 4. Solve the modified LP relaxation Results are good if the LP is a fair to good approximation of the MIP

Computational Challenge LP relaxation and rounding heuristic Compared to CPLEX 11.0 * feasible solution from later methods

Disjunctive Cutting Planes

LPCLVs As Disjunctive Programs m=5, K=2 x1x1 x2x2 b = 20 Disjunctive programming studies optimization problems over disjunctions (unions) of polyhedra (LP feasible sets) Easy to see how LPCLVs can be viewed this way as a union over all the LPs with different sets of (m-k) of the tighter b RHSs explicitly enforced (here all 3 of 5)

Valid Inequalities for Disjunctive Programming Balas proved that all needed valid inequalities for disjunctions of polyhedra can be constructed by rolling up each of the member linear systems with nonnegative multipliers, and picking the lowest of the rolled coefficients on the LHS and the largest on the RHS

Valid Inequalities for LPCLV – Main Disjunctive Inequality m=5, K=2 x1x1 x2x2 b = 20 Average of lowest 3 values Main Disjunctive Inequality

Valid Inequalities for LPCLV – A Family of Disjunctive Inequalities Lemma. For any subset S of i=1,…,m with K or more elements, the implied LPCLV(A[S],b,K) is a relaxation of the full LPCLV(A,b,K) where A[S] is the row sub- matrix of A for rows in S Proposition. For any subset S of i=1,…,m with more than K elements, the inequality is valid for full system LPCLV(A,b,K), where each is the average of the (|S|-K) smallest coefficients in rows in S.

Valid Inequalities for LPCLV – Disjunctive Support Inequalities m=5, K=2 x1x1 x2x2 b = 20 Average of the lowest value Disjunctive Support Inequality Proposition. Let subset S k of i=1,…,m be the K+1 rows with largest coefficients on x k. Then the inequality is valid for full system LPCLV(A,b,K), where each is smallest j coefficient for rows in S k. Furthermore the inequality supports the convex hull of solutions to the full LPCLV(A,b,K) (at x k = b /, other x j = 0)

Results for Disjunctive Cuts LP relaxation and rounding heuristic (except *) LP relaxation with disjunctive cuts and rounding heuristic * feasible solution from later methods

Single Constraint Enumeration

SCE Concept Consider relaxations enforcing only one row of the CLV system at a time

SCE Concept Proposition. Let z i denote the optimal solution value of SCE i. Then the k+1 st smallest such z i is a valid upper bound on the optimal solution value of the full LPCLV. (Proof idea: There can be at most k of the single constraints violated in LPCLV, but thereafter all constraints must be satisfied at the b value.) Row i zizi zizi zizi Example with m=9, k=3 Direct implementation solves all one-row problems, sorts solution values, and picks the k+1st Direct SCE (i) solves all one-row problems, (ii) sorts solution values, and (iii) picks the k+1 st smallest

Do better by using one-row problems solved to get adaptive upper/lower bounds for unsolved SCE i … … … Each Box Shows SCE Value Upper – Lower Range with Bar for True Optimal Value Solved Case K+1 st Lowest Upper Bound Is SCE Upper Bound K+1 st Lowest Lower Bound Is SCE Lower Bound Bounds Improved by Solving Other SCEs Adaptive Single Constraint Enumeration (ASCE)

ASCE – Global Bounds The K+1 st lowest of the upper bounds is a running upper bound on the ultimate SCE bound obtained (max of these upper bounds  max of their z’s >= max of ultimate K+1 z’s) The K+1 st lowest of the lower bounds is a running lower bound on the ultimate SCE bound obtained (max of these lower bounds <= max of lower bounds for ultimate K+1 <= max of their z’s)

ASCE – Surrogate Relaxation Upper Bounds Each SCE i solved produces dual multipliers that can be used to quickly update running upper bounds on solution values for other SCE q Construct a surrogate constraint valid for each SCE q using the dual solution for SCE i without any multipliers on the CLV constraints to roll up all the others Gx<=h For each q not i in turn, solve a simple relaxation of SCE q having only such surrogate constraints and the single CLV inequality for q These yield upper bounds on the solution values of the corresponding SCE q, and the running best upper bounds on their values can be updated

ASCE – Common Feasible for Lower Bounds If an SCE i solution vector is also feasible for constraint q of the CLV system, then i’s solution value is a lower bound on that SCE q and the running lower bound on that solution value can be updated

ASCE - Pruning If the running upper bound on the value of SCE q is already < the best known LPCLV feasible solution value, then q is certain to be among the lowest K+1 that establish the global bound. No further processing of SCE q is needed. If at least K+1 of the running upper bounds are no greater that the running lower bound for SCE q then it cannot be in the lowest K+1. No further exploration of q is required

Results with SCE and ASCE LP relaxation with disjunctive cuts and rounding heuristic LP relaxation with disjunctive cuts and SCE/ASCE and rounding heuristic

Conclusions Linear Programming Relaxations of MIP formulations of LPCLV problems can be arbitrarily weak Commercial MIP solver (CPLEX 11) is not effective in closing the initial large optimality gaps Efficient valid inequalities based on disjunctive programming theory can be generated to: Strengthen LP relaxations and significantly reduce initial optimality gaps when problems are dense Help find better feasible solutions within the rounding heuristic framework Improve run times

Conclusions SCE procedure can be implemented as a stand- alone procedure or in branch and bound Significantly reduces upper bound values. In some cases, finds better feasible solutions Adaptive SCE accomplishes the SCE computation in significantly reduced time The combined algorithm utilizing disjunctive inequalities and the ASCE procedure within the rounding heuristic framework produces lower optimality gaps and better feasible solution values