An interval algorithm for multi-objective optimization

Nonlinear biobjective optimization: improvements to interval branch & bound algorithms

“…Note that, if d v < d v , then the most distant point related to the vertex is in the envelope c y (e.g., see v (2) , v (3) , v (4) in the figure), otherwise the point maximizing the distance is in the left or the bottom bound of y (e.g., related to v (1) , the red triangle in the left bound of y maximize the distance; related to v (5) and v (6) , the red triangles in the bottom bound maximize the distance). (1) ← the vector in Y just before the first one dominated by y;…”

“…We compare the best strategy of the previous experiment (ref = inner-poly(20)) with the strategy using just the lower bound of c y (lower-c y ) and the strategy using both bounds of the constraint (full-c y ). For each strategy we considered the best value for the parameter N max in the set {5, 10, 20, 50, 100}, i.e., 20, 50 and 50 respectively 6 .…”

“…However, to our knowledge only a few exact methods have been proposed in literature for solving NLBOO problems (e.g., [6][7][8][9][10][11][12]). They are mainly based on interval arithmetic and branch & bound, and commonly find a thin envelope in the objective space, which certainly contains the set of non-dominated vectors Y * .…”

“…The Manhattan distance between two vectors (or points) a and b is defined as i |a i −b i | over the dimensions of the vectors 6. The larger value obtained for N max when c y is used, can be explained by the fact that, strategies using c y generally process a smaller number of boxes (in Table3, compare the number of processed boxes reported by each strategy), thus it is convenient to generate more feasible solutions in each box.…”

Simulation of Recursive Functions by Means of Interval Analysis and Pseudo-Orbits

“…While floating-point arithmetic is affected by rounding errors and can produce inaccurate results, interval arithmetic has the advantage of giving rigorous bounds for the exact solution. If the lower and upper bounds of the interval can be rounded down and rounded up, respectively, then finite precision calculations can be performed using intervals, to give an enclosure of the exact solution [7].…”