This is a prospective, multi-center, open-label, phase Ib/ II study (two parts) with patients that have locally advanced or metastatic HER2+ breast cancer. The first part (phase Ib) will investigate the MTD/ RP2D of the combination therapy of BEZ235 BID and weekly trastuzumab using a Bayesian model. Once MTD/ RP2D is established the second part (phase II) will start. Phase II will evaluate the efficacy and the safety of weekly trastuzumab plus BEZ235 BID compared to capecitabine and lapatinib.

Inclusion Criteria:

• Patient is a female ≥ 18 years of age
• Patient has a histologically and/or cytologically confirmed diagnosis of HER2-positive invasive breast cancer with inoperable locally advanced or metastatic disease
• Patients with controlled or asymptomatic CNS metastases are eligible
• Patient has adequate bone marrow and organ functions, and has recovery from all clinically significant toxicities related to prior anti-neoplastic therapies
• Absolute neutrophil count (ANC) ≥ 1.5 x 109/L
• Platelets ≥ 100 x 109/L
• Hemoglobin (Hgb) ≥ 9.0 g/dL
• INR ≤ 2
• Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤ 3 x ULN (or ≤ 5.0 x ULN if liver metastases are present)
• Total serum bilirubin ≤ 1.5 x ULN (in patients with known Gilbert Syndrome, a total bilirubin ≤ 3.0 x ULN, with direct bilirubin ≤ 1.5 x ULN)
• Serum creatinine ≤ 1.5 x ULN
• Fasting plasma glucose (FPG) ≤ 140mg/dL [7.8 mmol/L]
• HbA1c ≤ 8%
• Patient has received prior trastuzumab (alone or in combination) but NO more than 3 prior cytotoxic chemotherapy lines
• Prior endocrine and radiotherapy allowed
• Patient has ECOG performance status of 0-2 (Phase Ib) or 0-1 (Phase II)

Additional inclusion criteria for phase II:

• Available tumor tissue (archival or fresh) for biomarker analysis; known PI3K activation status
• At least one measurable lesion as per RECIST 1.1
• Patient has received prior treatment with a taxane
• Patient has "trastuzumab-resistance disease" defined as:
• Recurrence while on trastuzumab (or T-DM1) or within 12 months since the last infusion in the adjuvant setting
• Progression while on or within 4 weeks since the last infusion of trastuzumab (or T-DM1) in the locally advanced or metastatic setting

Exclusion Criteria:

• Previous treatment with PI3K and/or mTOR inhibitors
• Symptomatic/uncontrolled Central Nervous System (CNS) metastases
• Concurrent malignancy or malignancy in the last 3 years prior to enrollment
• Wide field radiotherapy ≤ 28 days or limited field radiation for palliation ≤ 14 days prior to starting study drug
• Active cardiac disease (e.g. LVEF less than institutional lower limit of normal, QTcF > 480 msec, unstable angina pectoris, ventricular, supraventricular or nodal arrhythmias)
• Inadequately controlled hypertension
• Impairment of gastrointestinal (GI) function or GI disease that may significantly alter the absorption of BEZ235
• Treatment at start of study treatment with drugs with a known risk to induce Torsades de Pointes, moderate and strong inhibitors or inducers of isoenzyme CYP3A4, warfarin and coumadin analogues, LHRH agonists
• Intolerance or contraindications to trastuzumab treatment
• Pregnant or nursing (lactating) woman

Additional exclusion criteria for phase II:

• Prior treatment with capecitabine and lapatinib
• Intolerance or contraindications to capecitabine and lapatinib
• Previous treatment with HER-2 targeted agents other than trastuzumab or T-DM1
• Peripheral neuropathy ≥ Grade 2